A334479 -id:A334479 - cp's OEIS frontend

A007528 Primes of the form 6k-1.

5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane

For values of k see A024898.
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
a(n) = A211890(3,n-1) for n <= 4. - Reinhard Zumkeller, Jul 13 2012
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
From Bernard Schott, Feb 14 2019: (Start)
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.)
{2,3} Union A002476 Union {this sequence} = A000040.
Except for 2 and 3, all Sophie Germain primes are of the form 6k-1.
Except for 3, all the lesser of twin primes are also of the form 6k-1.
Dirichlet's theorem on arithmetic progressions states that this sequence is infinite. (End)
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

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.
A. Mąkowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 125-126.
Wacław Sierpiński, Elementary Theory of Numbers, p. 144, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition, 1997, p. 127.

T. D. Noe, Table of n, a(n) for n = 1..1000
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].
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
Amelia Carolina Sparavigna, The Pentagonal Numbers and their Link to an Integer Sequence which contains the Primes of Form 6n-1, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Binary operations inspired by generalized entropies applied to figurate numbers, Politecnico di Torino (Italy, 2021).
Wikipedia, Dirichlet's theorem on arithmetic progressions.

Intersection of A016969 and A000040.
Cf. A003627, A010051, A117047, A132231, A214360, A057145.
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).
Cf. A034694 (smallest prime == 1 (mod n)).
Cf. A038700 (smallest prime == n-1 (mod n)).
Cf. A038026 (largest possible value of smallest prime == r (mod n)).
Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A048265, A324076.

Filtered(List([1..100],n->6*n-1),IsPrime); # Muniru A Asiru, May 19 2018

a007528 n = a007528_list !! (n-1)
a007528_list = [x | k <- [0..], let x = 6 * k + 5, a010051' x == 1]
-- Reinhard Zumkeller, Jul 13 2012

select(isprime,[seq(6*n-1,n=1..100)]); # Muniru A Asiru, May 19 2018

Select[6 Range[100]-1,PrimeQ]  (* Harvey P. Dale, Feb 14 2011 *)

forprime(p=2, 1e3, if(p%6==5, print1(p, ", "))) \\ Charles R Greathouse IV, Jul 15 2011

forprimestep(p=5,1000,6, print1(p", ")) \\ Charles R Greathouse IV, Mar 03 2025

A003627 \ {2}. - R. J. Mathar, Oct 28 2008
Conjecture: Product_{n >= 1} ((a(n) - 1) / (a(n) + 1)) * ((A002476(n) + 1) / (A002476(n) - 1)) = 3/4. - Dimitris Valianatos, Feb 11 2020
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 9*A175646/Pi^2 = 1/1.060548293.... =4/(3*A333240).
Product_{k>=1} (1 + 1/a(k)^2) = A334482.
Product_{k>=1} (1 - 1/a(k)^3) = A334480.
Product_{k>=1} (1 + 1/a(k)^3) = A334479. (End)
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

A334477 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^3).

Original entry on oeis.org

1, 0, 0, 3, 6, 0, 2, 5, 4, 0, 2, 2, 1, 2, 5, 9, 8, 9, 6, 7, 0, 4, 3, 2, 3, 9, 3, 3, 3, 3, 2, 1, 8, 7, 8, 5, 9, 1, 7, 0, 5, 3, 9, 4, 7, 7, 1, 1, 7, 5, 0, 8, 7, 2, 1, 3, 7, 0, 2, 2, 4, 0, 2, 6, 4, 1, 6, 5, 2, 3, 7, 1, 7, 3, 7, 1, 7, 3, 6, 2, 6, 1, 4, 6, 6, 2, 7, 5, 2, 0, 4, 0, 8, 1, 5, 1, 4, 8, 2, 9, 8, 9, 1, 5, 7

Offset: 1

Vaclav Kotesovec, May 02 2020

In general, for s > 0, Product_{k>=1} (1 + 1/A002476(k)^(2*s+1)) / (1 - 1/A002476(k)^(2*s+1)) = sqrt(3) * (2*Pi)^(2*s + 1) * zeta(2*s + 1) * A002114(s) / ((2^(2*s + 1) + 1) * (3^(2*s + 1) + 1) * (2*s)! * zeta(4*s + 2)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) / (1 - 1/A002476(k)^s) = (zeta(s, 1/6) - zeta(s, 5/6))*zeta(s) / ((2^s + 1)*(3^s + 1)*zeta(2*s)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) * (1 + 1/A007528(k)^s) = 6^s * zeta(s) / ((2^s + 1) * (3^s + 1) * zeta(2*s)).
For s > 0, Product_{k>=1} ((A007528(k)^(2*s+1) - 1) / (A007528(k)^(2*s+1) + 1)) * ((A002476(k)^(2*s+1) + 1) / (A002476(k)^(2*s+1) - 1)) = 6 * A002114(s)^2 * (4*s + 2)! / ((2^(4*s + 2) - 1) * (3^(4*s + 2) - 1) * Bernoulli(4*s + 2) * (2*s)!^2) = Bernoulli(2*s)^2 * (4*s + 2)! * (zeta(2*s + 1, 1/6) - zeta(2*s + 1, 5/6))^2 / (8*Pi^2 * (2^(4*s + 2) - 1) * (3^(4*s + 2) - 1) * Bernoulli(4*s + 2) * (2*s)!^2 * zeta(2*s)^2).

			1.0036025402212598967043239333321878591705394771...

Cf. A002476, A175646, A334424, A334426, A334478, A334481.

A334477 / A334478 = 15*sqrt(3)*zeta(3)/Pi^3.

A334477 * A334479 = 810*zeta(3)/Pi^6.

More digits from Vaclav Kotesovec, Jun 27 2020

A334480 Decimal expansion of Product_{k>=1} (1 - 1/A007528(k)^3).

Original entry on oeis.org

9, 9, 0, 8, 8, 4, 1, 4, 5, 5, 2, 5, 2, 1, 3, 3, 5, 6, 5, 6, 3, 4, 0, 3, 1, 7, 3, 5, 5, 9, 4, 3, 2, 7, 5, 1, 6, 4, 3, 4, 8, 3, 1, 2, 1, 7, 5, 0, 0, 7, 6, 1, 3, 3, 0, 4, 8, 6, 7, 7, 4, 7, 8, 4, 9, 4, 3, 1, 7, 8, 8, 8, 2, 5, 7, 6, 7, 4, 3, 1, 7, 7, 5, 2, 7, 6, 3, 4, 5, 2, 1, 7, 8, 9, 8, 8, 9, 2, 9, 2, 1, 3, 5, 4, 6, 7

Offset: 0

Vaclav Kotesovec, May 02 2020

In general, for s > 0, Product_{k>=1} (1 + 1/A007528(k)^(2*s+1)) / (1 - 1/A007528(k)^(2*s+1)) = (1 - 1/2^(2*s + 1)) * (3^(2*s + 1) - 1) * (2*s)! * zeta(2*s + 1) / (sqrt(3) * A002114(s) * Pi^(2*s + 1)).
For s > 1, Product_{k>=1} (1 + 1/A007528(k)^s) / (1 - 1/A007528(k)^s) = (2^s - 1) * (3^s - 1) * zeta(s) / (zeta(s, 1/6) - zeta(s, 5/6)).
For s > 1, Product_{k>=1} (1 - 1/A002476(k)^s) * (1 - 1/A007528(k)^s) = 6^s / ((2^s - 1)*(3^s - 1)*zeta(s)).

			0.990884145525213356563403173559432751643483121750... = 1/1.0091997177631243951237...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 6 5 3 = 1/A334480).

Cf. A007528, A175646, A334425, A334427, A334479.

A334479 / A334480 = 91*sqrt(3)*zeta(3)/(6*Pi^3).

A334478 * A334480 = 108/(91*zeta(3)).

More digits from Vaclav Kotesovec, Jun 27 2020

A334482 Decimal expansion of Product_{k>=1} (1 + 1/A007528(k)^2).

Original entry on oeis.org

1, 0, 5, 8, 7, 6, 0, 2, 0, 1, 7, 8, 2, 5, 4, 5, 4, 9, 1, 3, 1, 5, 8, 9, 5, 4, 5, 4, 5, 7, 2, 1, 5, 3, 3, 3, 6, 7, 3, 4, 7, 1, 2, 6, 6, 3, 2, 4, 9, 5, 1, 2, 2, 4, 0, 7, 9, 5, 9, 2, 7, 0, 1, 0, 8, 2, 2, 2, 9, 4, 1, 4, 4, 9, 9, 3, 8, 1, 9, 3, 0, 0, 7, 1, 8, 2, 1, 2, 7, 2, 3, 4, 9, 6, 3, 6, 0, 4, 8, 4, 2, 7, 2, 9, 7

Offset: 1

Vaclav Kotesovec, May 02 2020

Product_{k>=1} (1 - 1/A007528(k)^2) = 9*A175646/Pi^2 = 0.9429084997268899069451546585312672145658112624159...

			1.058760201782545491315895454572153336734712663249...

Cf. A007528, A175646, A334479, A334481.

A334481 * A334482 = 54/(5*Pi^2).

More digits from Vaclav Kotesovec, Jun 27 2020

A369566 Powerful numbers whose prime factors are all of the form 3*k + 2.

Original entry on oeis.org

1, 4, 8, 16, 25, 32, 64, 100, 121, 125, 128, 200, 256, 289, 400, 484, 500, 512, 529, 625, 800, 841, 968, 1000, 1024, 1156, 1331, 1600, 1681, 1936, 2000, 2048, 2116, 2209, 2312, 2500, 2809, 3025, 3125, 3200, 3364, 3481, 3872, 4000, 4096, 4232, 4624, 4913, 5000

Offset: 1

Amiram Eldar, Jan 26 2024

nonn

Closed under multiplication.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A004612.
Similar sequence: A352492, A369563, A369564, A369565.
Cf. A003627, A333240, A334479.

q[n_] := n == 1 || AllTrue[FactorInteger[n], Mod[First[#], 3] == 2 && Last[#] > 1 &]; Select[Range[5000], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 1]%3 != 2 || f[i, 2] == 1, return(0))); 1;}

Sum_{n>=1} 1/a(n) = Product_{primes p == 2 (mod 3)} (1 + 1/(p*(p-1))) = (9/8) * A333240 * A334479 = 1.6053538210...

A288143 Expansion of x * phi(x) * phi(x^3)^2 * f(x, x^5)^3 in powers of x where phi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 5, 9, 11, 24, 45, 50, 53, 81, 120, 120, 99, 170, 250, 216, 203, 288, 405, 362, 264, 450, 600, 528, 477, 601, 850, 729, 550, 840, 1080, 962, 821, 1080, 1440, 1200, 891, 1370, 1810, 1530, 1272, 1680, 2250, 1850, 1320, 1944, 2640, 2208, 1827, 2451, 3005, 2592

Offset: 1

Michael Somos, Jul 01 2017

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q + 5*q^2 + 9*q^3 + 11*q^4 + 24*q^5 + 45*q^6 + 50*q^7 + 53*q^8 + 81*q^9 + ...

Indranil Ghosh, Table of n, a(n) for n = 1..2000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A113261, A214262, A334478, A334479.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

A := Basis( ModularForms( Gamma1(12), 3), 52); A[2] + 5*A[3] + 9*A[4] + 11*A[5] + 24*A[6] + 45*A[7] + 50*A[8] + 53*A[9] + 81*A[10] + 120*A[11] + 120*A[12] + 99*A[13];

a[ n_] := If[ n < 1, 0, (-1)^n DivisorSum[ n, (-1)^# #^2 JacobiSymbol[ -3, n/#] &]];
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^3]^2 (QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6])^3, {x, 0, n}];
a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[# == 3, 9^#2, # == 2, (4^(#2 + 1) + 9 (-1)^(#2 + 1))/5, Mod[#, 6] == 1, ((#^2)^(#2 + 1) - 1)/(#^2 - 1), True, ((#^2)^(#2 + 1) - (-1)^(#2 + 1))/(#^2 + 1)] & @@@ FactorInteger@n)];

{a(n) = if( n<1, 0, (-1)^n * sumdiv( n, d, (-1)^d * d^2 * kronecker( -3, n/d)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^11 * eta(x^6 + A)^7 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A)^5 * eta(x^12 + A)), n))};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 9^e, p==2, (4^(e+1) + 9*(-1)^(e+1)) / 5, p%6==1, ((p^2)^(e+1) - 1) / (p^2 - 1), ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1))))};

Expansion of (a(q^2) - a(-q)) * (2*a(q) + a(-q))^2 / 54 in powers of q where a() is a cubic AGM theta function.
Expansion of -c(-q) * (2*c(q) + c(-q))^2 / 27 in powers of q where c() is a cubic AGM theta function.
Expansion of eta(q^2)^11 * eta(q^6)^7 / (eta(q)^5 * eta(q^3) * eta(q^4)^5 * eta(q^12)) in powers of q.
a(n) is multiplicative with a(3^e) = 9^e, a(2^e)  = (4^(e+1) + 9*(-1)^(e+1)) / 5 if e>0, a(p^e)  = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e)  = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 5 (mod 6).
Euler transform of period 12 sequence [5, -6, 6, -1, 5, -12, 5, -1, 6, -6, 5, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 192^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113261.
G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) * if(mod(k,4)=2, 3/2, 1).
a(n) = -(-1)^n * A214262(n).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime == 1 (mod 6)} (p^3/(p^3-1)) * Product_{p prime == 5 (mod 6)} (p^3/(p^3+1)) = 1/(A334478 * A334479) = 0.99452678821883983883... . - Amiram Eldar, Feb 20 2024

cp's OEIS Frontend

A007528 Primes of the form 6k-1.

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A334477 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^3).

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A334480 Decimal expansion of Product_{k>=1} (1 - 1/A007528(k)^3).

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A334482 Decimal expansion of Product_{k>=1} (1 + 1/A007528(k)^2).

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A369566 Powerful numbers whose prime factors are all of the form 3*k + 2.

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A288143 Expansion of x * phi(x) * phi(x^3)^2 * f(x, x^5)^3 in powers of x where phi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

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