A068229 -id:A068229 - cp's OEIS frontend

A317643 Expansion of theta_3(q^3)*theta_3(q^4), where theta_3() is the Jacobi theta function.

1, 0, 0, 2, 2, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of integer solutions to the equation 3*x^2 + 4*y^2 = n.

			G.f. = 1 + 2*q^3 + 2*q^4 + 4*q^7 + 2*q^12 + 6*q^16 + 4*q^19 + 2*q^27 + 4*q^28 + ...

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000049, A020677, A068229, A108563, A192323.

nmax = 100; CoefficientList[Series[EllipticTheta[3, 0, q^3] EllipticTheta[3, 0, q^4], {q, 0, nmax}], q]
nmax = 100; CoefficientList[Series[QPochhammer[-q^3, -q^3] QPochhammer[-q^4, -q^4]/(QPochhammer[q^3, -q^3] QPochhammer[q^4, -q^4]), {q, 0, nmax}], q]

G.f.: Product_{k>=1} (1 + x^(6*k-3))^2*(1 - x^(6*k))*(1 + x^(8*k-4))^2*(1 - x^(8*k)).

A358049 a(1) = 2, a(2) = 3; afterwards a(n) is least new prime > a(n-1) such that a(n-2) + a(n) and a(n-1) + a(n) are semiprimes.

Original entry on oeis.org

2, 3, 7, 19, 67, 127, 151, 271, 463, 823, 883, 991, 1051, 1087, 2011, 2251, 2311, 2371, 2383, 2731, 2803, 2971, 3271, 3391, 3643, 3823, 4111, 4483, 6343, 6379, 6763, 7879, 8443, 9199, 9283, 9643, 10159, 10639, 10867, 10939, 11047, 11299, 11467, 11587, 11971, 12511, 12583, 14071

Offset: 1

Zak Seidov, Oct 27 2022

nonn

Aside from the first two terms, all terms are 7 mod 12. - Charles R Greathouse IV, Dec 07 2022

			2 + 7 = 9 = 3*3 and 3 + 7 = 10 = 2*5 are semiprimes.

Cf. A001358.

Aside from the first two terms, a subsequence of A068229.

Do[While[MemberQ[s, p] || 2 != PrimeOmega[s[[-2]] + p] || 2 != PrimeOmega[s[[-1]] + p], p = NextPrime[p]]; AppendTo[s, p], {60}]; s

issp(n) = bigomega(n) == 2; \\ A001358
lista(nn) = my(va = vector(nn)); va[1] = 2; va[2] = 3; for (n=3, nn, my(p=nextprime(va[n-1]+1)); while (!issp(va[n-2]+p) || !issp(va[n-1]+p), p = nextprime(p+1)); va[n] = p;); va; \\ Michel Marcus, Nov 14 2022

A380877 Primes p where the prime race 12m+1 versus 12m+7 is tied.

Original entry on oeis.org

2, 3, 5, 13, 17, 433, 457, 461

Offset: 1

Ya-Ping Lu, Feb 06 2025

nonn

Primes p such that pi_{12,1}(p) = pi_{12,7}(p), where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m). For the first 5 billion primes, pi_{12,7}(p) >= pi_{12,1}(p). If exists, a(9) > 122430513841.

Cf. A007351, A068228, A068229, A379989, A380333.

s={};Do[p=Prime[pp];If[Length[Select[Prime[Range[pp]],Mod[#,12]==1&]]==Length[Select[Prime[Range[pp]],Mod[#,12]==7&]],AppendTo[s,p]],{pp,100}];s (* James C. McMahon, Mar 03 2025 *)

from sympy import nextprime; p, d = 2, 0
while p < 500:
    if d == 0: print(p, end = ', ')
    p = nextprime(p); r = p%12
    if r == 7: d += 1
    elif r == 1: d -= 1

cp's OEIS Frontend

A317643 Expansion of theta_3(q^3)*theta_3(q^4), where theta_3() is the Jacobi theta function.

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A358049 a(1) = 2, a(2) = 3; afterwards a(n) is least new prime > a(n-1) such that a(n-2) + a(n) and a(n-1) + a(n) are semiprimes.

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A380877 Primes p where the prime race 12m+1 versus 12m+7 is tied.

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Python