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%I M2476 N0982 #349 Feb 05 2024 00:52:36
%S 3,5,11,17,29,41,59,71,101,107,137,149,179,191,197,227,239,269,281,
%T 311,347,419,431,461,521,569,599,617,641,659,809,821,827,857,881,1019,
%U 1031,1049,1061,1091,1151,1229,1277,1289,1301,1319,1427,1451,1481,1487,1607
%N Lesser of twin primes.
%C Also, solutions to phi(n + 2) = sigma(n). - Conjectured by _Jud McCranie_, Jan 03 2001; proved by _Reinhard Zumkeller_, Dec 05 2002
%C 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
%C 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
%C It is conjectured that A113910(n+4) = a(n+2) for all n. - _Creighton Dement_, Jan 15 2009
%C 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
%C A164292(a(n)) = 1; A010051(a(n) - 2) = 0 for n > 1. - _Reinhard Zumkeller_, Mar 29 2010
%C From _Jonathan Sondow_, May 22 2010: (Start)
%C About 15% of primes < 19000 are the lesser of twin primes. About 26% of Ramanujan primes A104272 < 19000 are the lesser of twin primes.
%C About 46% of primes < 19000 are Ramanujan primes. About 78% of the lesser of twin primes < 19000 are Ramanujan primes.
%C 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)
%C Primes generated by sequence A040976. - _Odimar Fabeny_, Jul 12 2010
%C 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
%C 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
%C 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
%C Aside from the first term, all terms have digital root 2, 5, or 8. - _J. W. Helkenberg_, Jul 24 2013
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%C For a discussion of bias in the distribution of twin primes, see my article on the Vixra web site. - _Waldemar Puszkarz_, May 08 2018
%C 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
%C The yet unproved "Twin Prime Conjecture" states that this sequence is infinite. - _M. F. Hasler_, Nov 14 2019
%C 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
%C From _Lorenzo Sauras Altuzarra_, Dec 21 2021: (Start)
%C J. A. Hervás Contreras observed the subsequence 11, 311, 18311, 1518311, 421518311... (see the links), which led me to conjecture the following statements.
%C I. If i is an integer greater than 2, then there exist positive integers j and k such that a(j) equals the concatenation of 3k and a(i).
%C II. If k is a positive integer, then there exist positive integers i and j such that a(j) equals the concatenation of 3k and a(i).
%C III. If i, j, and r are positive integers such that i > 2 and a(j) equals the concatenation of r and a(i), then 3 divides r. (End)
%D 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.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 6.
%D P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1996, pp. 259-260.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Chris K. Caldwell, <a href="/A001359/b001359.txt">Table of n, a(n) for n = 1..100000</a>
%H Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Abhinav Aggarwal, Zekun Xu, Oluwaseyi Feyisetan, and Nathanael Teissier, <a href="https://arxiv.org/abs/2009.08559">On Primes, Log-Loss Scores and (No) Privacy</a>, arXiv:2009.08559 [cs.LG], 2020.
%H Chris K. Caldwell, <a href="http://www.utm.edu/research/primes/lists/small/100ktwins.txt">First 100000 Twin Primes</a>
%H Chris K. Caldwell, <a href="https://t5k.org/top20/page.php?id=1">Twin Primes</a>
%H Chris K. Caldwell, <a href="http://www.utm.edu/research/primes/largest.html#biggest">Largest known twin primes</a>
%H Chris K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=TwinPrime">Twin primes</a>
%H Chris K. Caldwell, <a href="http://www.utm.edu/research/primes/">The prime pages</a>
%H P. A. Clement, <a href="http://www.jstor.org/stable/2305816">Congruences for sets of primes</a>, American Mathematical Monthly, vol. 56,1 (1949), 23-25.
%H Harvey Dubner, <a href="http://www.emis.de/journals/JIS/VOL8/Dubner/dubner71.html">Twin Prime Statistics</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
%H Andrew Granville and Greg Martin, <a href="https://arxiv.org/abs/math/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004; Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
%H José Antonio Hervás Contreras, <a href="https://www.gaussianos.com/forogauss/topic/nueva-propiedad-de-los-primos-gemelos/">¿Nueva propiedad de los primos gemelos?</a>
%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/index.html">Some Results of Computational Research in Prime Numbers</a> [See local copy in A007053]
%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/twins/twins.html">Enumeration to 10^14 of the twin primes and Brun's constant</a>, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
%H Thomas R. Nicely, <a href="/A001359/a001359.pdf">Enumeration to 10^14 of the twin primes and Brun's constant</a> [Local copy, pdf only]
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H Waldemar Puszkarz, <a href="http://vixra.org/abs/1804.0416">Statistical Bias in the Distribution of Prime Pairs and Isolated Primes</a>, vixra:1804.0416 (2018).
%H Fred Richman, <a href="http://math.fau.edu/Richman/primes.htm">Generating primes by the sieve of Eratosthenes</a>
%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/1701.04741">New Congruences and Finite Difference Equations for Generalized Factorial Functions</a>, arXiv:1701.04741 [math.CO], 2017.
%H P. Shiu, <a href="http://dx.doi.org/10.1080/10586458.2005.10128903">A Diophantine Property Associated with Prime Twins</a>, Experimental mathematics 14 (1) (2005).
%H Jonathan Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, arXiv:0907.5232 [math.NT], 2009-2010; Amer. Math. Monthly, 116 (2009) 630-635.
%H Jonathan Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
%H Jonathan Sondow and Emmanuel Tsukerman, <a href="https://arxiv.org/abs/1401.0322">The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers</a>, arXiv:1401.0322 [math.NT], 2014; see section 4.
%H Terence Tao, <a href="https://arxiv.org/abs/math/0505402">Obstructions to uniformity and arithmetic patterns in the primes</a>, arXiv:math/0505402 [math.NT], 2005.
%H Apoloniusz Tyszka, <a href="https://philarchive.org/rec/TYSDAS">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)</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>
%F a(n) = A077800(2n-1).
%F A001359 = { n | A071538(n-1) = A071538(n)-1 }; A071538(A001359(n)) = n. - _M. F. Hasler_, Dec 10 2008
%F A001359 = { prime(n) : A069830(n) = A087454(n) }. - _Juri-Stepan Gerasimov_, Aug 23 2011
%F a(n) = prime(A029707(n)). - _R. J. Mathar_, Feb 19 2017
%p select(k->isprime(k+2),select(isprime,[$1..1616])); # _Peter Luschny_, Jul 21 2009
%p A001359 := proc(n)
%p option remember;
%p if n = 1
%p then 3;
%p else
%p p := nextprime(procname(n-1)) ;
%p while not isprime(p+2) do
%p p := nextprime(p) ;
%p end do:
%p p ;
%p end if;
%p end proc: # _R. J. Mathar_, Sep 03 2011
%t Select[Prime[Range[253]], PrimeQ[# + 2] &] (* _Robert G. Wilson v_, Jun 09 2005 *)
%t 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_ *)
%t 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 *)
%t Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&][[All,1]] (* _Harvey P. Dale_, Jan 04 2021 *)
%t q = Drop[Prepend[p = Prime[Range[100]], 2], -1];
%t Flatten[q[[#]] & /@ Position[p - q, 2]] (* _Horst H. Manninger_, Mar 28 2021 *)
%o (PARI) A001359(n,p=3) = { while( p+2 < (p=nextprime( p+1 )) || n-->0,); p-2}
%o /* The following gives a reasonably good estimate for any value of n from 1 to infinity; compare to A146214. */
%o 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)
%o /* The constant is A114907; the expression in front of +.5 is an estimate for A071538(x) */ \\ _M. F. Hasler_, Dec 10 2008
%o (Magma) [n: n in PrimesUpTo(1610) | IsPrime(n+2)]; // _Bruno Berselli_, Feb 28 2011
%o (Haskell)
%o a001359 n = a001359_list !! (n-1)
%o a001359_list = filter ((== 1) . a010051' . (+ 2)) a000040_list
%o -- _Reinhard Zumkeller_, Feb 10 2015
%o (Python)
%o from sympy import primerange, isprime
%o print([n for n in primerange(1, 2001) if isprime(n + 2)]) # _Indranil Ghosh_, Jul 20 2017
%Y Subsequence of A003627.
%Y Cf. A006512 (greater of twin primes), A014574, A001097, A077800, A002822, A040040, A054735, A067829, A082496, A088328, A117078, A117563, A074822, A071538, A007508, A146214, A350246, A350247.
%Y Cf. A104272 (Ramanujan primes), A178127 (lesser of twin Ramanujan primes), A178128 (lesser of twin primes if it is a Ramanujan prime).
%Y Cf. A010051, A000040.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
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