Search: helkenberg
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A001359
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Lesser of twin primes.
(Formerly M2476 N0982)
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+10
780
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3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Also, solutions to phi(n + 2) = sigma(n). - Conjectured by Jud McCranie, Jan 03 2001; proved by Reinhard Zumkeller, Dec 05 2002
The set of primes for which the weight as defined in A117078 is 3 gives this sequence except for the initial 3. - Rémi Eismann, Feb 15 2007
The set of lesser of twin primes larger than three is a proper subset of the set of primes of the form 3n - 1 (A003627). - Paul Muljadi, Jun 05 2008
It is conjectured that A113910(n+4) = a(n+2) for all n. - Creighton Dement, Jan 15 2009
I would like to conjecture that if f(x) is a series whose terms are x^n, where n represents the terms of sequence A001359, and if we inspect {f(x)}^5, the conjecture is that every term of the expansion, say a_n * x^n, where n is odd and at least equal to 15, has a_n >= 1. This is not true for {f(x)}^k, k = 1, 2, 3 or 4, but appears to be true for k >= 5. - Paul Bruckman (pbruckman(AT)hotmail.com), Feb 03 2009
A164292(a(n)) = 1; A010051(a(n) - 2) = 0 for n > 1. - Reinhard Zumkeller, Mar 29 2010
From Jonathan Sondow, May 22 2010: (Start)
About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes A104272 < 19000 are the lesser of twin primes.
About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan primes.
A reason for the jumps is in Section 7 of "Ramanujan primes and Bertrand's postulate" and in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps". (End)
Primes generated by sequence A040976. - Odimar Fabeny, Jul 12 2010
Primes of the form 2*n - 3 with 2*n - 1 prime n > 2. Primes of the form (n^2 - (n-2)^2)/2 - 1 with (n^2 - (n-2)^2)/2 + 1 prime so sum of two consecutive odd numbers/2 - 1. - Pierre CAMI, Jan 02 2012
Solutions of the equation n' + (n+2)' = 2, where n' is the arithmetic derivative of n. - Paolo P. Lava, Dec 18 2012
Conjecture: For any integers n >= m > 0, there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) (i.e., (a(m), ..., a(n)) in base b) is prime; moreover, when m = 1 there is such an integer b < (n+6)^2. - Zhi-Wei Sun, Mar 26 2013
Except for the initial 3, all terms are congruent to 5 mod 6. One consequence of this is that no term of this sequence appears in A030459. - Alonso del Arte, May 11 2013
Aside from the first term, all terms have digital root 2, 5, or 8. - _J. W. Helkenberg_, Jul 24 2013
The sequence provides all solutions to the generalized Winkler conjecture (A051451) aside from all multiples of 6. Specifically, these solutions start from n = 3 as a(n) - 3. This gives 8, 14, 26, 38, 56, ... An example from the conjecture is solution 38 from twin prime pairs (3, 5), (41, 43). - Bill McEachen, May 16 2014
Conjecture: a(n)^(1/n) is a strictly decreasing function of n. Namely a(n+1)^(1/(n+1)) < a(n)^(1/n) for all n. This conjecture is true for all a(n) <= 1121784847637957. - Jahangeer Kholdi and Farideh Firoozbakht, Nov 21 2014
a(n) are the only primes, p(j), such that (p(j+m) - p(j)) divides (p(j+m) + p(j)) for some m > 0, where p(j) = A000040(j). For all such cases m=1. It is easy to prove, for j > 1, the only common factor of (p(j+m) - p(j)) and (p(j+m) + p(j)) is 2, and there are no common factors if j = 1. Thus, p(j) and p(j+m) are twin primes. Also see A067829 which includes the prime 3. - Richard R. Forberg, Mar 25 2015
Primes prime(k) such that prime(k)! == 1 (mod prime(k+1)) with the exception of prime(991) = 7841 and other unknown primes prime(k) for which (prime(k)+1)*(prime(k)+2)*...*(prime(k+1)-2) == 1 (mod prime(k+1)) where prime(k+1) - prime(k) > 2. - Thomas Ordowski and Robert Israel, Jul 16 2016
For the twin prime criterion of Clement see the link. In Ribenboim, pp. 259-260 a more detailed proof is given. - Wolfdieter Lang, Oct 11 2017
Conjecture: Half of the twin prime pairs can be expressed as 8n + M where M > 8n and each value of M is a distinct composite integer with no more than two prime factors. For example, when n=1, M=21 as 8 + 21 = 29, the lesser of a twin prime pair. - Martin Michael Musatov, Dec 14 2017
For a discussion of bias in the distribution of twin primes, see my article on the Vixra web site. - Waldemar Puszkarz, May 08 2018
Since 2^p = 2 (mod p) (Fermat's little theorem), these are primes p such that 2^p == q (mod p), where q is the next prime after p. - Thomas Ordowski, Oct 29 2019, edited by M. F. Hasler, Nov 14 2019
The yet unproved "Twin Prime Conjecture" states that this sequence is infinite. - M. F. Hasler, Nov 14 2019
Lesser of the twin primes are the set of elements that occur in both A162566, A275697. Proof: A prime p will only have integer solutions to both (p+1)/g(p) and (p-1)/g(p) when p is the lesser of a twin prime, where g(p) is the gap between p and the next prime, because gcd(p+1,p-1) = 2. - Ryan Bresler, Feb 14 2021
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REFERENCES
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Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.
P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1996, pp. 259-260.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Chris K. Caldwell, Table of n, a(n) for n = 1..100000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Abhinav Aggarwal, Zekun Xu, Oluwaseyi Feyisetan, and Nathanael Teissier, On Primes, Log-Loss Scores and (No) Privacy, arXiv:2009.08559 [cs.LG], 2020.
Chris K. Caldwell, First 100000 Twin Primes
Chris K. Caldwell, Twin Primes
Chris K. Caldwell, Largest known twin primes
Chris K. Caldwell, Twin primes
Chris K. Caldwell, The prime pages
P. A. Clement, Congruences for sets of primes, American Mathematical Monthly, vol. 56,1 (1949), 23-25.
Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Thomas R. Nicely, Home page, which has extensive tables.
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Waldemar Puszkarz, Statistical Bias in the Distribution of Prime Pairs and Isolated Primes, vixra:1804.0416 (2018).
Fred Richman, Generating primes by the sieve of Eratosthenes
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
P. Shiu, A Diophantine Property Associated with Prime Twins, Experimental mathematics 14 (1) (2005).
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009) 630-635.
Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.
Jonathan Sondow and Emmanuel Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Terence Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Eric Weisstein's World of Mathematics, Twin Primes
Index entries for primes, gaps between
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FORMULA
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a(n) = A077800(2n-1).
A001359 = { n | A071538(n-1) = A071538(n)-1 } ; A071538(A001359(n)) = n. - M. F. Hasler, Dec 10 2008
A001359 = { prime(n) : A069830(n) = A087454(n) }. - Juri-Stepan Gerasimov, Aug 23 2011
a(n) = prime(A029707(n)). - R. J. Mathar, Feb 19 2017
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MAPLE
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select(k->isprime(k+2), select(isprime, [$1..1616])); # Peter Luschny, Jul 21 2009
A001359 := proc(n)
option remember;
if n = 1
then 3;
else
p := nextprime(procname(n-1)) ;
while not isprime(p+2) do
p := nextprime(p) ;
end do:
p ;
end if;
end proc: # R. J. Mathar, Sep 03 2011
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MATHEMATICA
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Select[Prime[Range[253]], PrimeQ[# + 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
a[n_] := a[n] = (p = NextPrime[a[n - 1]]; While[!PrimeQ[p + 2], p = NextPrime[p]]; p); a[1] = 3; Table[a[n], {n, 51}] (* Jean-François Alcover, Dec 13 2011, after R. J. Mathar *)
nextLesserTwinPrime[p_Integer] := Block[{q = p + 2}, While[NextPrime@ q - q > 2, q = NextPrime@ q]; q]; NestList[nextLesserTwinPrime@# &, 3, 50] (* Robert G. Wilson v, May 20 2014 *)
Select[Partition[Prime[Range[300]], 2, 1], #[[2]]-#[[1]]==2&][[All, 1]] (* Harvey P. Dale, Jan 04 2021 *)
q = Drop[Prepend[p = Prime[Range[100]], 2], -1];
Flatten[q[[#]] & /@ Position[p - q, 2]] (* Horst H. Manninger, Mar 28 2021 *)
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PROG
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(PARI) A001359(n, p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0, ); p-2}
/* The following gives a reasonably good estimate for any value of n from 1 to infinity; compare to A146214. */
A001359est(n) = solve( x=1, 5*n^2/log(n+1), 1.320323631693739*intnum(t=2.02, x+1/x, 1/log(t)^2)-log(x) +.5 - n)
/* The constant is A114907; the expression in front of +.5 is an estimate for A071538(x) */ \\ M. F. Hasler, Dec 10 2008
(MAGMA) [n: n in PrimesUpTo(1610) | IsPrime(n+2)]; // Bruno Berselli, Feb 28 2011
(Haskell)
a001359 n = a001359_list !! (n-1)
a001359_list = filter ((== 1) . a010051' . (+ 2)) a000040_list
-- Reinhard Zumkeller, Feb 10 2015
(Python)
from sympy import primerange, isprime
print([n for n in primerange(1, 2001) if isprime(n + 2)]) # Indranil Ghosh, Jul 20 2017
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CROSSREFS
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Subsequence of A003627.
Cf. A006512 (greater of twin primes), A014574, A001097, A077800, A002822, A040040, A054735, A067829, A082496, A088328, A117078, A117563, A074822, A071538, A007508, A146214.
Cf. A104272 (Ramanujan primes), A178127 (lesser of twin Ramanujan primes), A178128 (lesser of twin primes if it is a Ramanujan prime).
Cf. A010051, A000040.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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A006512
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Greater of twin primes.
(Formerly M3763)
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+10
417
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5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Also primes that are the sum of two primes (which is possible only if 2 is one of the primes). - Cino Hilliard, Jul 02 2004, edited by M. F. Hasler, Nov 14 2019
The set of greater of twin primes larger than five is a proper subset of the set of primes of the form 3n + 1 (A002476). - Paul Muljadi, Jun 05 2008
Smallest prime > n-th isolated composite. - Juri-Stepan Gerasimov, Nov 07 2009
Subsequence of A175075. Union of a(n) and sequence A175080 is A175075. - Jaroslav Krizek, Jan 30 2010
A164292(a(n))=1; A010051(a(n)+2)=0 for n > 1. - Reinhard Zumkeller, Mar 29 2010
Omega(n) = Omega(n-2); d(n) = d(n-2). - Juri-Stepan Gerasimov, Sep 19 2010
Solutions of the equation (n-2)'+n' = 2, where n' is the arithmetic derivative of n. - Paolo P. Lava, Dec 18 2012
Aside from the first term, all subsequent terms have digital root 1, 4, or 7. - _J. W. Helkenberg_, Jul 24 2013
Also primes p with property that the sum of the successive gaps between primes <= p is a prime number. - Robert G. Wilson v, Dec 19 2014
The phrase "x is an element of the {primes, positive integers} and there {exist no, exist} elements a,b of {1 and primes, primes}: a+b=x" determines A133410, A067829, A025584, A006512, A166081, A014092, A014091 and A038609 for the first few hundred terms with only de-duplication or omitting/including 3, 4 and 6 in the case of A166081/A014091 and one case of omitting/including 3 given 1 isn't prime. - Harry G. Coin, Nov 25 2015
The yet unproved Twin Prime Conjecture states that this sequence is infinite. - M. F. Hasler, Nov 14 2019
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REFERENCES
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See A001359 for further references and links.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
R. K. Guy, Letter to N. J. A. Sloane, Jun 1991
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Wikipedia, Twin prime.
Index entries for primes, gaps between
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MAPLE
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for i from 1 to 253 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007
P := select(isprime, [$1..1609]): select(p->member(p-2, P), P); # Peter Luschny, Mar 03 2011
A006512 := proc(n)
2+A001359(n) ;
end proc: # R. J. Mathar, Nov 26 2014
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MATHEMATICA
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Select[Prime[Range[254]], PrimeQ[# - 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
Transpose[Select[Partition[Prime[Range[300]], 2, 1], Last[#] - First[#] == 2 &]][[2]] (* Harvey P. Dale, Nov 02 2011 *)
Cases[Prime[Range[500]] + 2, _?PrimeQ] (* Fred Patrick Doty, Aug 23 2017 *)
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PROG
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(PARI) select(p->isprime(p-2), primes(1000))
(MAGMA) [n: n in PrimesUpTo(1610)|IsPrime(n-2)]; // Bruno Berselli, Feb 28 2011
(Haskell)
a006512 = (+ 2) . a001359 -- Reinhard Zumkeller, Feb 10 2015
(PARI) a(n)=p=3; while(p+2 < (p=nextprime(p+1)) || n-->0, ); p
vector(100, n, a(n)) \\ Altug Alkan, Dec 04 2015
(Python)
from sympy import primerange, isprime
print([n for n in primerange(1, 2001) if isprime(n - 2)]) # Indranil Ghosh, Jul 20 2017
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CROSSREFS
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Subsequence of A139690.
Bisection of A077800.
Cf. A001097, A001359, A014574, A067829, A002476.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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A014574
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Average of twin prime pairs.
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+10
333
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4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1608
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graph;
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listen;
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text;
internal format)
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OFFSET
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1,1
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COMMENTS
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With an initial 1 added, this is the complement of the closure of {2} under a*b+1 and a*b-1. - Franklin T. Adams-Watters, Jan 11 2006
Also the square root of the product of twin prime pairs + 1. Two consecutive odd numbers can be written as 2k+1,2k+3. Then (2k+1)(2k+3)+1 = 4(k^2+2k+1) = 4(k+1)^2, a perfect square. Since twin prime pairs are two consecutive odd numbers, the statement is true for all twin prime pairs. - Cino Hilliard, May 03 2006
Or, single (or isolated) composites. Nonprimes k such that neither k-1 nor k+1 is nonprime. - Juri-Stepan Gerasimov, Aug 11 2009
Numbers n such that sigma(n-1) = phi(n+1). - Farideh Firoozbakht, Jul 04 2010
Solutions of the equation (n-1)'+(n+1)'=2, where n' is the arithmetic derivative of n. - Paolo P. Lava, Dec 18 2012
Aside from the first term in the sequence, all remaining terms have digital root 3, 6, or 9. - _J. W. Helkenberg_, Jul 24 2013
Numbers n such that n^2-1 is a semiprime. - Thomas Ordowski, Sep 24 2015
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REFERENCES
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Archimedeans Problems Drive, Eureka, 30 (1967).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Glossary: Twin primes
C. K. Caldwell, The Top Twenty: Twin Primes
Y. Fujiwara, Parsing a Sequence of Qubits, IEEE Trans. Information Theory, 59 (2013), 6796-6806.
Y. Fujiwara, Parsing a Sequence of Qubits, arXiv:1207.1138 [quant-ph], 2012-2013.
L. J. Gerstein, A reformulation of the Goldbach conjecture, Math. Mag., 66 (1993), 44-45.
Eric Weisstein's World of Mathematics, Twin Primes
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FORMULA
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a(n) = (A001359(n) + A006512(n))/2 = 2*A040040(n) = A054735(n)/2 = A111046(n)/4.
a(n) = A129297(n+4). - Reinhard Zumkeller, Apr 09 2007
A010051(a(n) - 1) * A010051(a(n) + 1) = 1. Reinhard Zumkeller, Apr 11 2012
a(n) = 6*A002822(n-1), n>=2. - Ivan N. Ianakiev, Aug 19 2013
a(n)^4 - 4*a(n)^2 = A062354(a(n)^2 - 1). - Raphie Frank, Oct 17 2013
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MAPLE
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P := select(isprime, [$1..1609]): map(p->p+1, select(p->member(p+2, P), P)); # Peter Luschny, Mar 03 2011
A014574 := proc(n) option remember; local p ; if n = 1 then 4 ; else p := nextprime( procname(n-1) ) ; while not isprime(p+2) do p := nextprime(p) ; od ; return p+1 ; end if ; end proc: # R. J. Mathar, Jun 11 2011
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MATHEMATICA
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Select[Table[Prime[n] + 1, {n, 260}], PrimeQ[ # + 1] &] (* Ray Chandler, Oct 12 2005 *)
Mean/@Select[Partition[Prime[Range[300]], 2, 1], Last[#]-First[#]==2&] (* Harvey P. Dale, Jan 16 2014 *)
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PROG
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(PARI) p=2; forprime(q=3, 1e4, if(q-p==2, print1(p+1", ")); p=q) \\ Charles R Greathouse IV, Jun 10 2011
(Maxima) A014574(n) := block(
if n = 1 then
return(4),
p : A014574(n-1) ,
for k : 2 step 2 do (
if primep(p+k-1) and primep(p+k+1) then
return(p+k)
)
)$ /* R. J. Mathar, Mar 15 2012 */
(Haskell)
a014574 n = a014574_list !! (n-1)
a014574_list = [x | x <- [2, 4..], a010051 (x-1) == 1, a010051 (x+1) == 1]
-- Reinhard Zumkeller, Apr 11 2012
(GAP) a:=1+Filtered([1..2000], p->IsPrime(p) and IsPrime(p+2)); # Muniru A Asiru, May 20 2018
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CROSSREFS
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Cf. A000010, A000203, A001359, A002822, A006512, A037074, A040040, A054735, A077800, A111046.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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R. K. Guy, N. J. A. Sloane, Eric W. Weisstein
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EXTENSIONS
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Offset changed to 1 by R. J. Mathar, Jun 11 2011
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STATUS
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approved
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A040040
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Average of twin prime pairs (A014574), divided by 2. Equivalently, 2*a(n)-1 and 2*a(n)+1 are primes.
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+10
42
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2, 3, 6, 9, 15, 21, 30, 36, 51, 54, 69, 75, 90, 96, 99, 114, 120, 135, 141, 156, 174, 210, 216, 231, 261, 285, 300, 309, 321, 330, 405, 411, 414, 429, 441, 510, 516, 525, 531, 546, 576, 615, 639, 645, 651, 660, 714, 726, 741, 744, 804, 810, 834, 849, 861, 894
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Intersection of A005097 and A006254. - Zak Seidov, Mar 18 2005
The only possible pairs for 2a(n)+/-1 are prime/prime (this sequence), not prime/not prime (A104278), prime/notprime (A104279) and not prime/prime (A104280), ... this sequence + A104280 + A104279 + A104278 = the odd numbers.
These numbers are never k mod (2k+1) or (k+1) mod (2k+1) with 2k+1 < a(n). - Jon Perry, Sep 04 2012
Excluding the first term, all remaining terms have digital root 3, 6 or 9. - _J. W. Helkenberg_, Jul 24 2013
Positive numbers x such that the difference between x^2 and adjacent squares are prime (both x^2-(x-1)^2 and (x+1)^2-x^2 are prime). - Doug Bell, Aug 21 2015
A260689(a(n),1) = A264526(a(n)) = 1. - Reinhard Zumkeller, Nov 17 2015
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10001
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FORMULA
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a(n) = A014574(n)/2 = A054735(n+1)/4 = A111046(n+1)/8.
For n>1, a(n) = 3*A002822(n-1). - Jason Kimberley, Nov 06 2015
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MAPLE
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P := select(isprime, [$1..1789]): map(p->(p+1)/2, select(p->member(p+2, P), P)); # Peter Luschny, Mar 03 2011
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MATHEMATICA
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Select[Range[900], And @@ PrimeQ[{-1, 1} + 2# ] &] (* Ray Chandler, Oct 12 2005 *)
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PROG
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(PARI) p=2; forprime(b=3, 1e4, if(b-p==2, print1((p+1)/2", ")); p=b) \\ Altug Alkan, Nov 10 2015
(Haskell)
a040040 = flip div 2 . a014574 -- Reinhard Zumkeller, Nov 17 2015
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CROSSREFS
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Cf. A001359, A006512, A014574, A054735, A111046, A045753 (even terms halved), A002822 (terms divided by 3).
Cf. A221310.
Cf. A260689, A264526.
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Cino Hilliard, Oct 21 2002
Title corrected by Daniel Forgues, Jun 01 2009
Edited by Daniel Forgues, Jun 21 2009
Comment corrected by Daniel Forgues, Jul 12 2009
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STATUS
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approved
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A054735
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Sum of twin prime pairs.
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+10
39
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8, 12, 24, 36, 60, 84, 120, 144, 204, 216, 276, 300, 360, 384, 396, 456, 480, 540, 564, 624, 696, 840, 864, 924, 1044, 1140, 1200, 1236, 1284, 1320, 1620, 1644, 1656, 1716, 1764, 2040, 2064, 2100, 2124, 2184, 2304, 2460, 2556, 2580, 2604, 2640, 2856, 2904
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OFFSET
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1,1
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COMMENTS
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(p^q)+(q^p) calculated modulo pq, where (p,q) is the n-th twin prime pair. Example: (599^601)+(601^599) == 1200 mod (599*601). - Sam Alexander, Nov 14 2003
El'hakk makes the following claim (without any proof): (q^p)+(p^q) = 2*cosh(q arctanh( sqrt( 1-((2/p)^2) ) )) + 2cosh(p arctanh( sqrt( 1-((2/q)^2) ) )) mod p*q. - Sam Alexander, Nov 14 2003
Also: Numbers N such that N/2-1 and N/2+1 both are prime. - M. F. Hasler, Jan 03 2013
Excluding the first term, all remaining terms have digital root 3, 6 or 9. - _J. W. Helkenberg_, Jul 24 2013
Except for the first term, this sequence is a subsequence of A005101 (Abundant numbers) and of A008594 (Multiples of 12). - Ivan N. Ianakiev, Jul 04 2021
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
El'hakk, Page of the time traveler [Archived copy on web.archive.org, as of Oct 28 2009.]
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FORMULA
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a(n) = 2*A014574(n) = 4*A040040(n) = A111046(n)/2.
a(n) = 12*A002822(n-1) for all n > 1. - M. F. Hasler, Dec 12 2019
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EXAMPLE
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a(3) = 24 because the twin primes 11 and 13 add to 24.
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MAPLE
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ZL:=[]:for p from 1 to 1451 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL), p+(p+2)]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007
A054735 := proc(n)
2*A001359(n)+2;
end proc: # R. J. Mathar, Jan 06 2013
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MATHEMATICA
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Select[Table[Prime[n] + 1, {n, 230}], PrimeQ[ # + 1] &] *2 (* Ray Chandler, Oct 12 2005 *)
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PROG
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(PARI) is_A054735(n)={!bittest(n, 0)&&isprime(n\2-1)&&isprime(n\2+1)} \\ M. F. Hasler, Jan 03 2013
(PARI) pp=1; forprime(p=1, 1482, if( p==pp+2, print1(p+pp, ", ")); pp=p) \\ Following a suggestion by R. J. Cano, Jan 05 2013
(Haskell)
a054735 = (+ 2) . (* 2) . a001359 -- Reinhard Zumkeller, Feb 10 2015
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CROSSREFS
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Cf. A001359, A006512, A014574, A040040, A111046.
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KEYWORD
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easy,nonn
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AUTHOR
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Enoch Haga, Apr 22 2000
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EXTENSIONS
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Additional comments from Ray Chandler, Nov 16 2003
Broken link fixed by M. F. Hasler, Jan 03 2013
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STATUS
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approved
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A181732
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Numbers n such that 90n + 1 is prime.
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+10
26
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2, 3, 6, 7, 9, 11, 13, 17, 18, 20, 24, 25, 26, 28, 31, 33, 34, 37, 39, 41, 45, 47, 51, 54, 55, 62, 65, 68, 69, 70, 72, 73, 74, 76, 84, 86, 89, 90, 91, 94, 96, 97, 98, 100, 101, 102, 107, 108, 109, 110, 117, 119, 121, 123, 124, 125, 130, 133, 136, 138, 139, 140
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OFFSET
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1,1
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = (1/90) [A142312(n) - 1].
a(n) ~ (4/15) n log n. - Charles R Greathouse IV, Jun 01 2016, corrected Sep 05 2016
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MAPLE
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a:= proc(n) option remember; local k;
for k from 1+ `if`(n=1, 1, a(n-1))
while not isprime(90*k+1) do od; k
end:
seq(a(n), n=1..100); # Alois P. Heinz, Dec 05 2011
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MATHEMATICA
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Select[Range[200], PrimeQ[90 # + 1] &] (* Vincenzo Librandi, Sep 06 2016 *)
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PROG
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(PARI) is(n)=n%90==1 && isprime(n) \\ Charles R Greathouse IV, Jun 01 2016
(MAGMA) [n: n in [0..200]| IsPrime(90*n+1)]; // Vincenzo Librandi, Sep 06 2016
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CROSSREFS
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Cf. A142312. Complement of A255491.
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KEYWORD
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nonn,easy
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AUTHOR
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_J. W. Helkenberg_, Nov 16 2010
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EXTENSIONS
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New name from Michael B. Porter, Jul 26 2013
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STATUS
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approved
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A198382
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Numbers n such that 90n + 37 is prime.
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+10
22
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0, 1, 3, 4, 5, 6, 8, 10, 12, 14, 17, 18, 19, 22, 25, 26, 27, 28, 29, 32, 35, 38, 39, 40, 41, 43, 46, 48, 49, 55, 56, 57, 59, 60, 61, 67, 69, 70, 71, 73, 77, 78, 80, 82, 83, 85, 87, 92, 96, 101, 104, 116, 117, 118, 120, 124, 125, 127, 131, 133, 134, 136, 138
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OFFSET
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1,3
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LINKS
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Ivan Neretin, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = (A142331(n)-37)/90.
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MATHEMATICA
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Select[Range[0, 150], PrimeQ[90 # + 37] &] (* Vincenzo Librandi, Apr 07 2015 *)
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PROG
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(PARI) isok(n) = isprime(90*n+37) \\ Michel Marcus, Jul 24 2013
(MAGMA) [n: n in [0..150] | IsPrime(90*n+37)]; // Vincenzo Librandi, Apr 07 2015
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CROSSREFS
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Cf. A181732.
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KEYWORD
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nonn,easy
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AUTHOR
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_J. W. Helkenberg_, Oct 24 2011
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STATUS
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approved
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A195993
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Numbers n such that 90n + 73 is prime.
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+10
21
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0, 1, 4, 5, 6, 9, 11, 12, 15, 18, 19, 20, 22, 23, 27, 28, 29, 32, 36, 39, 40, 42, 43, 49, 51, 54, 55, 56, 61, 62, 63, 65, 70, 72, 74, 75, 85, 88, 91, 92, 93, 95, 96, 97, 98, 103, 104, 106, 109, 110, 113, 114, 116, 127, 128, 131
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OFFSET
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1,3
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COMMENTS
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This sequence results from the propagation (addition) of 12 Fibonacci-like sequences; this sequence contains (recovers) all digital root 1 and last digit 3 prime numbers.
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LINKS
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Ivan Neretin, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = (A142326(n)-73)/90.
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MAPLE
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A142326 := proc(n)
option remember;
if n = 1 then
73 ;
else
a := nextprime(procname(n-1)) ;
while (a mod 45) <> 28 do
a := nextprime(a) ;
end do;
return a;
end if;
end proc:
A195993 := proc(n)
(A142326(n)-73)/90 ;
end proc:
seq(A195993(n), n=1..80) ; # R. J. Mathar, Oct 31 2011
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MATHEMATICA
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Select[Range[0, 200], PrimeQ[90#+73]&] (* Harvey P. Dale, May 05 2014 *)
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PROG
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(PARI) is(n)=isprime(90*n+73) \\ Charles R Greathouse IV, Apr 25 2016
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CROSSREFS
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Cf. A181732, A198382.
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KEYWORD
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nonn,easy
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AUTHOR
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_J. W. Helkenberg_, Oct 27 2011
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STATUS
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approved
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A196000
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Numbers k such that 90*k + 19 is prime.
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+10
21
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0, 1, 2, 4, 8, 9, 10, 11, 14, 16, 17, 22, 23, 24, 25, 28, 30, 34, 35, 36, 39, 41, 43, 46, 48, 50, 53, 55, 56, 60, 63, 64, 65, 69, 74, 77, 78, 79, 80, 81, 83, 85, 86, 91, 93, 98, 99, 101, 102, 107, 108, 109, 111, 112, 115, 116
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OFFSET
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1,3
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COMMENTS
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A142322 is a digital root 1 and last digit 9 preserving sequence.
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LINKS
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Ivan Neretin, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = (A142322(n) - 19)/90.
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MAPLE
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A142322 := proc(n)
option remember;
if n = 1 then
19 ;
else
a := nextprime(procname(n-1)) ;
while (a mod 45) <> 19 do
a := nextprime(a) ;
end do;
return a;
end if;
end proc:
A196000 := proc(n)
(A142322(n)-19)/90 ;
end proc:
seq(A196000(n), n=1..80) ; # R. J. Mathar, Oct 31 2011
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MATHEMATICA
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Select[Range[0, 120], PrimeQ[90 # + 19] &] (* Ivan Neretin, Apr 27 2017 *)
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PROG
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(PARI) is(n)=isprime(90*n+19) \\ Charles R Greathouse IV, Apr 25 2016
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CROSSREFS
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Cf. A195953, A181732, A198382.
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KEYWORD
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nonn,easy
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AUTHOR
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_J. W. Helkenberg_, Oct 27 2011
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STATUS
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approved
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A063983
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Least k such that k*2^n +/- 1 are twin primes.
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+10
19
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4, 2, 1, 9, 12, 6, 3, 9, 57, 30, 15, 99, 165, 90, 45, 24, 12, 6, 3, 69, 132, 66, 33, 486, 243, 324, 162, 81, 90, 45, 345, 681, 585, 375, 267, 426, 213, 429, 288, 144, 72, 36, 18, 9, 147, 810, 405, 354, 177, 1854, 927, 1125, 1197, 666, 333, 519, 1032, 516, 258, 129, 72
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OFFSET
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0,1
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COMMENTS
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Excluding the first three terms, all remaining terms have digital root 3, 6, or 9. - _J. W. Helkenberg_, Jul 24 2013
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REFERENCES
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Richard Crandall and Carl Pomerance, 'Prime Numbers: A Computational Perspective,' Springer-Verlag, NY, 2001, page 12.
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LINKS
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Pierre CAMI, Table of n, a(n) for n = 0..2300
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EXAMPLE
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a(3) = 9 because 9*2^3=72 and 71 and 73 are twin primes.
n=6: a(6)=3, 64.3=192 and {191,193} are both primes; n=71: a(71)=630, 630*[2^71]=1487545442103938242314240 and {1487545442103938242314239, 1487545442103938242314241} are twin primes.
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MATHEMATICA
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Table[Do[s=(2^j)*k; If[PrimeQ[s-1]&&PrimeQ[s+1], Print[{j, k]], {k, 1, 2*j^2], {j, 0, 100]; (*outprint of a[j]=k*)
Do[ k = 1; While[ ! PrimeQ[ k*2^n + 1 ] || ! PrimeQ[ k*2^n - 1 ], k++ ]; Print[ k ], {n, 0, 50} ]
f[n_] := Block[{k = 1}, While[Nand @@ PrimeQ[{-1, 1} + 2^n*k], k++ ]; k]; Table[f[n], {n, 60}] (* Ray Chandler, Jan 09 2009 *)
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CROSSREFS
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Cf. A040040, A045753, A002822, A124065, A124518-A124522.
Cf. A071256, A060210, A060256. For records see A125848, A125019.
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KEYWORD
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nonn
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AUTHOR
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Robert G. Wilson v, Sep 06 2001
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EXTENSIONS
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More terms from Labos Elemer, May 24 2002
Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar
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STATUS
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approved
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