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A001359
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Lesser of twin primes.
(Formerly M2476 N0982)
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393
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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
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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
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 (creighton.k.dement(AT)uni-oldenburg.de), 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]
Largest prime < n-th isolated composite. [Juri-Stepan Gerasimov, Nov 07 2009]
A164292(a(n)) = 1; A010051(a(n)-2) = 0 for n > 1. [Reinhard Zumkeller, Mar 29 2010]
Contribution 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
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REFERENCES
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M. Abramowitz and I. 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.
Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
T. R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
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).
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
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LINKS
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C. K. Caldwell, Table of n, a(n) for n = 1..100000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. K. Caldwell, First 100000 Twin Primes
C. K. Caldwell, Twin Primes
C. K. Caldwell, Largest known twin primes
C. K. Caldwell, Twin primes
C. K. Caldwell, The prime pages
A. Granville and G. Martin, Prime number races
Thomas R. Nicely, Home page, which has extensive tables.
O. E. Pol, Determinacion geometrica de los numeros primos y perfectos.
F. Richman, Generating primes by the sieve of Eratosthenes
P. Shiu, A Diophantine Property Associated with Prime Twins, Experimental mathematics 14 (1) (2005)
T. Tao, Obstructions to uniformity and arithmetic patterns in the primes
Eric Weisstein's World of Mathematics, Twin Primes
Index entries for primes, gaps between
J. Sondow, Ramanujan primes and Bertrand's postulate
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2
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FORMULA
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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]
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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] &] (* from 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, 1, 51}] (* Jean-François Alcover, Dec 13 2011, after R. J. Mathar *)
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PROG
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Contribution from M. F. Hasler, Dec 10 2008: (Start)
(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) */ (End)
(MAGMA) [n: n in PrimesUpTo(1610) | IsPrime(n+2)]; // Bruno Berselli, Feb 28 2011
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CROSSREFS
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Cf. A006512 (greater of twin primes), A014574, A001097, A077800.
a(n) = A077800(2n-1).
Cf. A002822, A040040, A054735, A067829, A082496, A088328, A117078, A117563, A001359, A074822, A003627.
Cf. A071538, A007508, A146214.
Cf. A104272 Ramanujan primes, A178127 Lesser of twin Ramanujan primes, A178128 Lesser of twin primes if it is a Ramanujan prime.
Sequence in context: A093326 * A096292 A181747 A078864 A208574 A023218
Adjacent sequences: A001356 A001357 A001358 * A001360 A001361 A001362
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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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