A334478 -id:A334478

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A002476

Primes of the form 6m + 1.
(Formerly M4344 N1819)

+10
263

7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Equivalently, primes of the form 3m + 1.

Rational primes that decompose in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017

Primes p dividing Sum_{k=0..p} binomial(2k, k) - 3 = A006134(p) - 3. - Benoit Cloitre, Feb 08 2003

Primes p such that tau(p) == 2 (mod 3) where tau(x) is the Ramanujan tau function (cf. A000594). - Benoit Cloitre, May 04 2003

Primes of the form x^2 + xy - 2y^2 = (x+2y)(x-y). - N. J. A. Sloane, May 31 2014

Primes of the form x^2 - xy + 7y^2 with x and y nonnegative. - T. D. Noe, May 07 2005

Primes p such that p^2 divides Sum_{m=1..2(p-1)} Sum_{k=1..m} (2k)!/(k!)^2. - Alexander Adamchuk, Jul 04 2006

A006512 larger than 5 (Greater of twin primes) is a subsequence of this. - Jonathan Vos Post, Sep 03 2006

A039701(A049084(a(n))) = A134323(A049084(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2007

Also primes p such that the arithmetic mean of divisors of p^2 is an integer: sigma_1(p^2)/sigma_0(p^2) = C. (A000203(p^2)/A000005(p^2) = C). - Ctibor O. Zizka, Sep 15 2008

Fermat knew that these numbers can also be expressed as x^2 + 3y^2 and are therefore not prime in Z[omega], where omega is a complex cubic root of unity. - Alonso del Arte, Dec 07 2012

Primes of the form x^2 + xy + y^2 with x < y and nonnegative. Also see A007645 which also applies when x=y, adding an initial 3. - Richard R. Forberg, Apr 11 2016

For any term p in this sequence, let k = (p^2 - 1)/6; then A016921(k) = p^2. - Sergey Pavlov, Dec 16 2016; corrected Dec 18 2016

For the decomposition p=x^2+3*y^2, x(n) = A001479(n+1) and y(n) = A001480(n+1). - R. J. Mathar, Apr 16 2024

REFERENCES

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.

David A. Cox, Primes of the Form x^2 + ny^2. New York: Wiley (1989): 8.

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

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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. Banderier, Calcul de (-3/p)

Barry Brent, Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups, Integers (2024) Vol. 24, Art. No. A18. See p. 13.

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.

A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.

Neville Robbins, On the Infinitude of Primes of the Form 3k+1, Fib. Q., 43,1 (2005), 29-30.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Index to sequences related to decomposition of primes in quadratic fields

FORMULA

From R. J. Mathar, Apr 03 2011: (Start)

Sum_{n >= 1} 1/a(n)^2 = A175644.

Sum_{n >= 1} 1/a(n)^3 = A175645. (End)

a(n) = 6*A024899(n) + 1. - Zak Seidov, Aug 31 2016

From Vaclav Kotesovec, May 02 2020: (Start)

Product_{k>=1} (1 - 1/a(k)^2) = 1/A175646.

Product_{k>=1} (1 + 1/a(k)^2) = A334481.

Product_{k>=1} (1 - 1/a(k)^3) = A334478.

Product_{k>=1} (1 + 1/a(k)^3) = A334477. (End)

Legendre symbol (-3, a(n)) = +1 and (-3, A007528(n)) = -1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

EXAMPLE

Since 6 * 1 + 1 = 7 and 7 is prime, 7 is in the sequence. (Also 7 = 2^2 + 3 * 1^2 = (2 + sqrt(-3))(2 - sqrt(-3)).)

Since 6 * 2 + 1 = 13 and 13 is prime, 13 is in the sequence.

17 is prime but it is of the form 6m - 1 rather than 6m + 1, and is therefore not in the sequence.

MAPLE

a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];

MATHEMATICA

Select[6*Range[100] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 06 2006 *)

PROG

(Magma) [n: n in [1..700 by 6] | IsPrime(n)]; // Vincenzo Librandi, Apr 05 2011

(PARI) select(p->p%3==1, primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

(Haskell)

a002476 n = a002476_list !! (n-1)

a002476_list = filter ((== 1) . (`mod` 6)) a000040_list

-- Reinhard Zumkeller, Jan 15 2013

(J) (#~ 1&p:) >: 6 * i.1000 NB. Stephen Makdisi, May 01 2018

(GAP) Filtered(List([0..110], k->6*k+1), n-> IsPrime(n)); # Muniru A Asiru, Mar 11 2019

CROSSREFS

Cf. A045331, A242660.

For values of m see A024899. Primes of form 3n - 1 give A003627.

These are the primes arising in A024892, A024899, A034936.

A091178 gives prime index.

Cf. A006512, A007528.

Subsequence of A016921 and of A050931.

Cf. A004611 (multiplicative closure).

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A175646

Decimal expansion of the Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2).

+10
31

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The Euler product of the Riemann zeta function at 2 restricted to primes in A002476, which is the inverse of the infinite product (1-1/7^2)*(1-1/13^2)*(1-1/19^2)*...

There is a complementary Product_{primes p == 2 (mod 3)} 1/(1-1/p^2) = A333240 = 1.4140643908921476375655018190798... such that (this constant here)*1.4140643.../(1-1/3^2) = zeta(2) = A013661.

Because 1/(1-p^(-2)) = 1+1/(p^2-1), the complementary 1.414064... also equals Product_{primes p == 2 (mod 3)} (1+1/(p^2-1)), which appears in Eq. (1.8) of [Dence and Pomerance]. - R. J. Mathar, Jan 31 2013

LINKS

Peter Luschny, Table of n, a(n) for n = 1..1000 (terms 1..105 from Vaclav Kotesovec).

Thomas Dence and Carl Pomerance, Euler's Function in Residue Classes, Raman. J., Vol. 2 (1998) pp. 7-20, alternative link.

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.

FORMULA

Equals 2*Pi^2 / (3^(7/2) * A301429^2). - Vaclav Kotesovec, May 12 2020

Equals Sum_{k>=1} 1/A004611(k)^2. - Amiram Eldar, Sep 27 2020

EXAMPLE

1.03401487541434188053903064441304762857896...

MAPLE

z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):

x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n) / 3:

evalf(4*Pi^2 / (27*mul(x(n), n=1..8)), 106); # Peter Luschny, Jan 17 2021

MATHEMATICA

digits = 105;

precision = digits + 5;

prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];

Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;

Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];

Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];

gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];

pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;

pA = Pi^2/9/pB ;

RealDigits[pA, 10, digits][[1]]

(* Jean-François Alcover, Jan 11 2021, after PARI code due to Artur Jasinski *)

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);

P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];

Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);

$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[3, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

z[n_] := Zeta[n] / Im[PolyLog[n, (-1)^(2/3)]];

x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n) / 3;

N[4 Pi^2 / (27 Product[x[n], {n, 8}]), 106] (* Peter Luschny, Jan 17 2021 *)

CROSSREFS

Cf. A002476, A004611, A007528, A175647, A301429, A333240, A334478, A334480.

KEYWORD

cons,nonn

AUTHOR

R. J. Mathar, Aug 01 2010

EXTENSIONS

More digits from Vaclav Kotesovec, May 12 2020 and Jun 27 2020

STATUS

approved

A334477

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

+10
6

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..105.

FORMULA

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

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

EXAMPLE

1.0036025402212598967043239333321878591705394771...

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, May 02 2020

EXTENSIONS

More digits from Vaclav Kotesovec, Jun 27 2020

STATUS

approved

A334480

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

+10
5

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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)).

LINKS

Table of n, a(n) for n=0..105.

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).

FORMULA

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

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

EXAMPLE

0.990884145525213356563403173559432751643483121750... = 1/1.0091997177631243951237...

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, May 02 2020

EXTENSIONS

More digits from Vaclav Kotesovec, Jun 27 2020

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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).

LINKS

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.

FORMULA

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

EXAMPLE

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 + ...

MATHEMATICA

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)];

PROG

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

(PARI) {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))};

(PARI) {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))))};

(Magma) 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];

CROSSREFS

Cf. A113261, A214262, A334478, A334479.

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

KEYWORD

nonn,mult

AUTHOR

Michael Somos, Jul 01 2017

STATUS

approved

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