A245643 -id:A245643 - cp's OEIS frontend

A245639 Prime numbers P such that 8*P^2-1 is also prime.

2, 3, 5, 11, 17, 19, 23, 31, 59, 67, 79, 89, 103, 107, 137, 173, 193, 229, 233, 241, 257, 263, 271, 311, 317, 353, 359, 383, 409, 431, 479, 509, 521, 523, 541, 563, 569, 577, 593, 599, 613, 641, 709, 739, 751, 787, 829, 887, 907, 919, 947, 971, 983, 1033

Offset: 1

Pierre CAMI, Jul 28 2014

			8*2^2-1=31 prime so a(1)=2.
8*3^2-1=71 prime so a(2)=3.
8*5^2-1=199 prime so a(3)=5.
8*7^2-1=391 composite.
8*11^2-1=967 prime so a(4)=11.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A245640, A245641, A245642, A245643.

[p: p in PrimesUpTo(1500)| IsPrime(8*p^2-1)]; // Vincenzo Librandi, Sep 07 2014

Reap[Do[p = Prime[n]; If[PrimeQ[8*p^2-1], Sow[p]], {n, 1, 200}]][[2, 1]] (* Jean-François Alcover, Jul 28 2014 *)
Select[Prime[Range[200]], PrimeQ[8 #^2 - 1] &] (* Vincenzo Librandi, Sep 07 2014 *)

select(p->isprime(8*p^2-1), primes(300)) \\ Colin Barker, Jul 28 2014

import sympy
from sympy import isprime
from sympy import prime
for n in range(1,10**3):
  p = prime(n)
  if isprime(8*p**2-1):
    print(p,end=', ')
# Derek Orr, Aug 13 2014

A134416 Expansion of eta(q^4)^2 / (eta(q^2) * eta(q)^6) in powers of q.

Original entry on oeis.org

1, 6, 28, 104, 342, 1016, 2808, 7296, 18044, 42750, 97656, 215992, 464360, 973176, 1993328, 3998592, 7870038, 15221232, 28968084, 54311736, 100421688, 183281904, 330468216, 589084288, 1038850488, 1813500030, 3135518440, 5372110496, 9124793472, 15371832424

Offset: 0

Michael Somos, Oct 26 2007

nonn

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

			G.f. = 1 + 6*q + 28*q^2 + 104*q^3 + 342*q^4 + 1016*q^5 + 2808*q^6 + 7296*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A134414, A245643.

nmax = 40; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k)^5, {k, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ 1 / (EllipticTheta[ 4, 0 , q]^3 EllipticTheta[ 4, 0, q^2]^2), {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)
a[ n_] := SeriesCoefficient[ 1 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q]^4), {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 / EllipticTheta[ 4, 0, q^2]^8, {q, 0, n}]; (* Michael Somos, Oct 16 2015 *)
QP = QPochhammer; s = QP[q^4]^2/(QP[q^2]*QP[q]^6) + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x^2 + A) * eta(x + A)^6), n))};

q='q+O('q^99); Vec(eta(q^4)^2/(eta(q^2)*eta(q)^6)) \\ Altug Alkan, Apr 16 2018

Euler transform of period 4 sequence [ 6, 7, 6, 5, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k)^5. [corrected by Vaclav Kotesovec, Sep 07 2015]
a(n) ~ exp(2*Pi*sqrt(n))/(32*n^2). - Vaclav Kotesovec, Sep 07 2015
-2 * a(n) = A134414(4*n).
Expansion of psi(q^2) / f(-q)^6 = phi(q)^3 / phi(-q^2)^8 = 1 / (phi(-q)^3 * phi(-q^2)^2) = 1 / (phi(q) * phi(-q)^4) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Oct 16 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(-13/2) (t/i)^(-5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134414. - Michael Somos, Oct 16 2015
Convolution inverse is A245643. - Michael Somos, Oct 16 2015

A263398 Expansion of phi(-x^2)^6 * psi(x^6) / f(x)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, -7, 14, 20, -36, -34, 40, 50, -30, -71, 76, 82, -144, -98, 112, 131, -70, -140, 170, 168, -288, -228, 232, 246, -120, -290, 258, 310, -468, -280, 344, 337, -190, -350, 394, 412, -648, -510, 496, 462, -252, -583, 558, 602, -864, -532, 584, 664, -350

Offset: 0

Michael Somos, Oct 16 2015

sign

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

			G.f. = 1 - 2*x - 7*x^2 + 14*x^3 + 20*x^4 - 36*x^5 - 34*x^6 + 40*x^7 + ...
G.f. = q^2 - 2*q^5 - 7*q^8 + 14*q^11 + 20*q^14 - 36*q^17 - 34*q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A245643.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2]^6 EllipticTheta[ 2, 0, x^3] / (2 x^(3/4) QPochhammer[ -x]^2), {x, 0, n}];

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

q='q+O('q^99); Vec(eta(q)^2*eta(q^2)^6*eta(q^12)^2/(eta(q^4)^4*eta(q^6))) \\ Altug Alkan, Jul 31 2018

Expansion of q^(-2/3) * eta(q)^2 * eta(q^2)^6 * eta(q^12)^2 / (eta(q^4)^4 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [-2, -8, -2, -4, -2, -7, -2, -4, -2, -8, -2, -5, ...].
8 * a(n) = A245643(3*n + 2).

A266575 Expansion of q * f(-q^4)^6 / phi(-q) in powers of q where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 4, 8, 8, 12, 16, 16, 25, 28, 28, 32, 40, 40, 48, 64, 48, 62, 76, 64, 80, 92, 80, 96, 121, 100, 112, 128, 120, 136, 160, 128, 144, 184, 152, 200, 200, 164, 208, 224, 192, 216, 252, 224, 248, 296, 224, 256, 337, 262, 312, 320, 280, 336, 368, 320, 336, 396

Offset: 1

Michael Somos, Jan 03 2016

nonn

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

			G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 16*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A134414. A245643.

A := Basis( ModularForms( Gamma1(4), 5/2), 59); A[2];

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^4]^6 / EllipticTheta[ 4, 0, q], {q, 0, n}];
a[ n_] := SeriesCoefficient[ 2^-4 EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^4, {q, 0, n}];

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

Expansion of q * phi(q) * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^4)^6 / eta(q)^2 in powers of q.
Euler transform of period 4 sequence [2, 1, 2, -5, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(-3/2) (t/I)^(5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A245643.
G.f.: x * Product_{k>0} (1 + x^k) * (1 - x^(4*k))^6 / (1 - x^k).
Convolution inverse of A134414.

A242113 a(n) = number of primes of the form k^n - m^k where k > m > 0.

Original entry on oeis.org

0, 1, 2, 6, 7, 2, 14, 7, 11, 10, 33, 10, 42, 35, 47, 39, 122, 22, 248, 113, 247, 236, 751, 75, 1268, 812, 1422, 1531, 4543, 87, 8669, 5750, 8884, 10983, 29084, 2274, 58841, 41242, 58030, 74646, 216647, 11656, 419147, 313237, 364925, 617742, 1576642, 75542, 3071839, 2299620

Offset: 1

Juri-Stepan Gerasimov, Aug 15 2014

nonn

It would be good to have a proof that a(n) is always finite. - N. J. A. Sloane, Sep 06 2014

			a(2) = 1 because  2^2 - 1^2 = 3 is prime;
a(3) = 2 because  2^3 - 1^2 = 7 is prime and 3^3 - 2^3 = 19 is prime, but 2^3 - 2^3 < 0, 5^3 - 2^5 = 93 is not prime, 5^3 - 2^7 = 215 is not prime, 9^3 - 2^9 = 217 is not prime, 11^3 - 2^11 < 0.
More generally, primes of the form k^r - m^k where  k > m > 0:
r = 2: 3;
r = 3: 7, 19;
r = 4: 7, 17, 73, 593, 2273, 20369;
r = 5: 7, 23, 31, 179, 58537, 1951811, 1986949;
r = 6: 4818617, 24006497;
r = 7: 7, 47, 79, 103, 127, 1137, 2179, 77101, 162287, 543607, 1706527, 9940951, 6069961193, 25365130463;
r = 8: 31, 6553, 141793, 49046209, 815722529, 16983038753, 499709542049;
r = 9: 71, 151, 223, 431, 463, 487, 503, 4521799, 133227103, 10604491181, 1175888158183;
r = 10: 4177, 37097, 58049, 58537, 1803001, 2486784401, 3486783889, 41426502825041, 819626139497153, 52458394747474721.

Cf. A045575, A123206, A245459, A245643.

f[r_] := Length@ Rest@ Union@ Flatten@ Table[ If[ PrimeQ[k^r - m^k], k^r - m^k, 0], {k, 2, 10000000}, {m, Floor[k^(r/k)]}]; Do[ Print[ f[r]], {r, 2, 50}] (* Robert G. Wilson v, Aug 25 2014 *)

a(n) >= A245459(n).

a(10)-a(50) from Robert G. Wilson v, Aug 25 2014

A246608 Expansion of phi(-q) * phi(-q^4)^4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, -6, 16, 0, 0, 8, -50, 0, 0, 16, 80, 0, 0, -38, -96, 0, 0, -16, 160, 0, 0, 48, -242, 0, 0, 64, 240, 0, 0, -56, -288, 0, 0, -150, 400, 0, 0, 112, -384, 0, 0, 112, 496, 0, 0, -112, -674, 0, 0, -80, 560, 0, 0, 160, -672, 0, 0, 192, 880, 0, 0, -294

Offset: 0

Michael Somos, Sep 01 2014

sign

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

			G.f. = 1 - 2*q - 6*q^4 + 16*q^5 + 8*q^8 - 50*q^9 + 16*q^12 + 80*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A244276, A245643.

A := Basis( ModularForms( Gamma0(8), 5/2), 68); A[1] - 2*A[2];

a[n_]:= SeriesCoefficient[EllipticTheta[3,0, -q]*EllipticTheta[3,0, -q^4 ]^4, {q, 0, n}]; (* corrected by G. C. Greubel, Mar 15 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^8 / (eta(x^2 + A) * eta(x^8 + A)^4), n))};

Expansion of eta(q)^2 * eta(q^4)^8 / (eta(q^2) * eta(q^8)^4) in powers of q.

a(4*n) = A245643(n). a(4*n + 1) = -2 * A244276(n). a(4*n + 2) = a(4*n + 3) = 0.

cp's OEIS Frontend

A245639 Prime numbers P such that 8*P^2-1 is also prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A134416 Expansion of eta(q^4)^2 / (eta(q^2) * eta(q)^6) in powers of q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A263398 Expansion of phi(-x^2)^6 * psi(x^6) / f(x)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A266575 Expansion of q * f(-q^4)^6 / phi(-q) in powers of q where phi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A242113 a(n) = number of primes of the form k^n - m^k where k > m > 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A246608 Expansion of phi(-q) * phi(-q^4)^4 in powers of q where phi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula