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 A001359 Lesser of twin primes. (Formerly M2476 N0982) 810
 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)
 OFFSET 1,1 COMMENTS 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 From Lorenzo Sauras Altuzarra, Dec 21 2021: (Start) J. A. Hervás Contreras observed the subsequence 11, 311, 18311, 1518311, 421518311... (see the links), which led me to conjecture the following statements. 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). 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). 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) REFERENCES 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). LINKS 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, arXiv:math/0408319 [math.NT], 2004; Amer. Math. Monthly, 113 (No. 1, 2006), 1-33. José Antonio Hervás Contreras, ¿Nueva propiedad de los primos gemelos? Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053] 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. Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant [Local copy, pdf only] 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, arXiv:0907.5232 [math.NT], 2009-2010; Amer. Math. Monthly, 116 (2009) 630-635. Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; 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. Eric Weisstein's World of Mathematics, Twin Primes FORMULA 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 MAPLE 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 MATHEMATICA Select[Prime[Range], 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 = 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], 2, 1], #[]-#[]==2&][[All, 1]] (* Harvey P. Dale, Jan 04 2021 *) q = Drop[Prepend[p = Prime[Range], 2], -1]; Flatten[q[[#]] & /@ Position[p - q, 2]] (* Horst H. Manninger, Mar 28 2021 *) PROG (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 CROSSREFS Subsequence of A003627. Cf. A006512 (greater of twin primes), A014574, A001097, A077800, A002822, A040040, A054735, A067829, A082496, A088328, A117078, A117563, A074822, A071538, A007508, A146214, A350246, A350247. Cf. A104272 (Ramanujan primes), A178127 (lesser of twin Ramanujan primes), A178128 (lesser of twin primes if it is a Ramanujan prime). Cf. A010051, A000040. Sequence in context: A078859 A054799 A093326 * A096292 A181747 A078864 Adjacent sequences:  A001356 A001357 A001358 * A001360 A001361 A001362 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified September 27 10:36 EDT 2022. Contains 357057 sequences. (Running on oeis4.)