cp's OEIS Frontend

A007528 Primes of the form 6k-1.

%I A007528 M3809 #158 Mar 03 2025 09:00:53
%S A007528 5,11,17,23,29,41,47,53,59,71,83,89,101,107,113,131,137,149,167,173,
%T A007528 179,191,197,227,233,239,251,257,263,269,281,293,311,317,347,353,359,
%U A007528 383,389,401,419,431,443,449,461,467,479,491,503,509,521,557,563,569,587
%N A007528 Primes of the form 6k-1.
%C A007528 For values of k see A024898.
%C A007528 Also primes p such that p^q - 2 is not prime where q is an odd prime. These numbers cannot be prime because the binomial p^q = (6k-1)^q expands to 6h-1 some h. Then p^q-2 = 6h-1-2 is divisible by 3 thus not prime. - _Cino Hilliard_, Nov 12 2008
%C A007528 a(n) = A211890(3,n-1) for n <= 4. - _Reinhard Zumkeller_, Jul 13 2012
%C A007528 There exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - _Ralf Steiner_, May 17 2018
%C A007528 From _Bernard Schott_, Feb 14 2019: (Start)
%C A007528 A theorem due to Andrzej Mąkowski: every integer greater than 161 is the sum of distinct primes of the form 6k-1. Examples: 162 = 5 + 11 + 17 + 23 + 47 + 59; 163 = 17 + 23 + 29 + 41 + 53. (See Sierpiński and David Wells.)
%C A007528 {2,3} Union A002476 Union {this sequence} = A000040.
%C A007528 Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
%C A007528 Except for 3, all the lesser of twin primes are also of the form 6k-1.
%C A007528 Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
%C A007528 For all elements of this sequence p=6*k-1, there are no (x,y) positive integers such that k=6*x*y-x+y. - _Pedro Caceres_, Apr 06 2019
%D A007528 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.
%D A007528 A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
%D A007528 Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
%D A007528 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007528 David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.
%H A007528 T. D. Noe, <a href="/A007528/b007528.txt">Table of n, a(n) for n = 1..1000</a>
%H A007528 M. Abramowitz and I. 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 A007528 F. S. Carey, <a href="https://archive.org/stream/proceedingslond00unkngoog#page/n317/mode/2up">On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p</a>, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
%H A007528 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.4641886">The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1</a>, Politecnico di Torino (Italy, 2021).
%H A007528 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.4662489">Binary operations inspired by generalized entropies applied to figurate numbers</a>, Politecnico di Torino (Italy, 2021).
%H A007528 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions">Dirichlet's theorem on arithmetic progressions</a>.
%F A007528 A003627 \ {2}. - _R. J. Mathar_, Oct 28 2008
%F A007528 Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - _Dimitris Valianatos_, Feb 11 2020
%F A007528 From _Vaclav Kotesovec_, May 02 2020: (Start)
%F A007528 Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
%F A007528 Product_{k>=1} (1 + 1/a(k)^2) = A334482.
%F A007528 Product_{k>=1} (1 - 1/a(k)^3) = A334480.
%F A007528 Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
%F A007528 Legendre symbol (-3, a(n)) = -1 and (-3, A002476(n)) = +1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - _Wolfdieter Lang_, Mar 03 2021
%p A007528 select(isprime,[seq(6*n-1,n=1..100)]); # _Muniru A Asiru_, May 19 2018
%t A007528 Select[6 Range[100]-1,PrimeQ]  (* _Harvey P. Dale_, Feb 14 2011 *)
%o A007528 (PARI) forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ _Charles R Greathouse IV_, Jul 15 2011
%o A007528 (PARI) forprimestep(p=5,1000,6, print1(p", ")) \\ _Charles R Greathouse IV_, Mar 03 2025
%o A007528 (Haskell)
%o A007528 a007528 n = a007528_list !! (n-1)
%o A007528 a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
%o A007528 -- _Reinhard Zumkeller_, Jul 13 2012
%o A007528 (GAP) Filtered(List([1..100],n->6*n-1),IsPrime); # _Muniru A Asiru_, May 19 2018
%Y A007528 Intersection of A016969 and A000040.
%Y A007528 Cf. A003627, A010051, A117047, A132231, A214360, A057145.
%Y A007528 Prime sequences A# (k,r) of the form k*n+r with 0 <= r <= k-1 (i.e., primes == r (mod k), or primes p with p mod k = r) and gcd(r,k)=1: A000040 (1,0), A065091 (2,1), A002476 (3,1), A003627 (3,2), A002144 (4,1), A002145 (4,3), A030430 (5,1), A045380 (5,2), A030431 (5,3), A030433 (5,4), A002476 (6,1), this sequence (6,5), A140444 (7,1), A045392 (7,2), A045437 (7,3), A045471 (7,4), A045458 (7,5), A045473 (7,6), A007519 (8,1), A007520 (8,3), A007521 (8,5), A007522 (8,7), A061237 (9,1), A061238 (9,2), A061239 (9,4), A061240 (9,5), A061241 (9,7), A061242 (9,8), A030430 (10,1), A030431 (10,3), A030432 (10,7), A030433 (10,9), A141849 (11,1), A090187 (11,2), A141850 (11,3), A141851 (11,4), A141852 (11,5), A141853 (11,6), A141854 (11,7), A141855 (11,8), A141856 (11,9), A141857 (11,10), A068228 (12,1), A040117 (12,5), A068229 (12,7), A068231 (12,11).
%Y A007528 Cf. A034694 (smallest prime == 1 (mod n)).
%Y A007528 Cf. A038700 (smallest prime == n-1 (mod n)).
%Y A007528 Cf. A038026 (largest possible value of smallest prime == r (mod n)).
%Y A007528 Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
%Y A007528 Cf. A048265, A324076.
%K A007528 nonn,easy
%O A007528 1,1
%A A007528 _N. J. A. Sloane_