A224609 -id:A224609 - cp's OEIS frontend

A224610 Smallest j such that j2prime(n)^3-1 and j2prime(n)*q^2-1 are prime.

2, 2, 5, 7, 59, 142, 264, 25, 8, 21, 124, 33, 60, 87, 9, 231, 5, 6, 82, 155, 7, 66, 72, 21, 42, 105, 15, 48, 250, 68, 222, 54, 47, 195, 255, 360, 205, 6, 83, 26, 5, 1, 50, 220, 173, 1, 976, 30, 228, 130, 30, 129, 46, 1106, 65, 62, 15, 109, 24, 41, 922, 15, 132, 89

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

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

Cf. A224490, A224609, A224611, A224612.

a[n_] := For[p = Prime[n]; j = 1, j < 10^6, j++, If[PrimeQ[q = j*2*p^3 - 1] && PrimeQ[j*p*2*q^2 - 1], Return[j]]]; Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Apr 22 2013 *)

A224611 Smallest j such that j2p(n)^3-1=q is prime, j2p(n)q^2-1=r, j2p(n)r^2-1=s, where r and s are also prime.

Original entry on oeis.org

902, 145, 771, 1060, 3569, 520, 938, 294, 2457, 3911, 1650, 483, 8604, 3450, 2345, 548, 25004, 1635, 5767, 14519, 2518, 6394, 198, 7961, 4272, 8370, 4146, 654, 4489, 6987, 222, 5426, 5250, 17670, 7691, 360, 3994, 20821, 9008, 6525, 9204, 1464, 6111, 6625, 11229, 3315, 62340, 735, 6962, 5236

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

Pierre CAMI, Table of n, a(n) for n = 1..3500

Cf. A224491, A224609, A224610, A224612.

a[n_] := For[j = 1, j < 10^7, j++, p = Prime[n]; If[PrimeQ[q = j*2*p^3 - 1] && PrimeQ[r = j*2*p*q^2 - 1] && PrimeQ[j*2*p*r^2 - 1], Return[j]]]; Table[ Print[an = a[n]]; an, {n, 1, 50}] (* Jean-François Alcover, Apr 12 2013 *)

A224612 Let p = prime(n). Smallest j such that q = j2p^3-1, r = jp2q^2-1, s = jp2r^2-1, and jp2*s^2-1 are prime numbers.

Original entry on oeis.org

29952, 12063, 1463, 6102, 11661, 49552, 639179, 2099290, 291248, 393186, 545251, 321303, 436641, 278295, 746832, 237852, 56490, 165901, 152847, 619755, 777177, 3410085, 117513, 2015421, 497170, 14750, 161190, 347039, 251835, 57536, 222, 2083286, 384944, 1228474, 3909960, 344164, 332224, 207751, 14060

Offset: 1

Pierre CAMI, Apr 12 2013

nonn

Conjecture: a(n) exists for all n.

Cf. A224492, A224609, A224610, A224611.

f:= proc(n) local j,p,q,r,s;
    p:= ithprime(n);
    for j from 1 do
      q:= j*2*p^3-1; if not isprime(q) then next fi;
      r:= j*p*2*q^2-1; if not isprime(r) then next fi;
      s:= j*p*2*r^2-1; if not isprime(s) then next fi;
      if isprime(j*p*2*s^2-1) then return j fi;
    od
end proc;
map(f, [$1..25]); # Robert Israel, May 15 2025

a[n_] := For[j = 1, j < 10^7, j++, p = Prime[n]; If[PrimeQ[q = j*2*p^3 - 1] && PrimeQ[r = j*2*p*q^2 - 1] && PrimeQ[s = j*2*p*r^2 - 1] && PrimeQ[j*2*p*s^2 - 1], Return[j]]]; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Apr 12 2013 *)

More terms from Jean-François Alcover, Apr 12 2013

Name clarified and more terms from Robert Israel, May 15 2025

A255258 Expansion of q^2 * phi(q) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0

Offset: 2

Michael Somos, Feb 19 2015

nonn

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

			G.f. = q^2 + 2*q^3 + 2*q^6 + 2*q^11 + 3*q^18 + 2*q^19 + 2*q^22 + 4*q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 2..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033761, A113411, A224609, A227395.

A := Basis( ModularForms( Gamma1(32), 1), 89); A[3] + 2*A[4] + 2*A[7] + 2*A[12];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^8] / 2, {q, 0, n}];

{a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^32 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)), n))};

Expansion of eta(q^2)^5 * eta(q^32)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^16)) in powers of q.
Euler transform of period 32 sequence [ 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A224609.
(-1)^n * a(n) = A227395(n).
a(4*n) = a(4*n + 1) = a(8*n + 7) = 0. a(4*n + 2) = A113411(n). a(8*n + 3) = 2 * A033761(n).

cp's OEIS Frontend

A224610 Smallest j such that j2prime(n)^3-1 and j2prime(n)*q^2-1 are prime.

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A224611 Smallest j such that j2p(n)^3-1=q is prime, j2p(n)q^2-1=r, j2p(n)r^2-1=s, where r and s are also prime.

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A224612 Let p = prime(n). Smallest j such that q = j2p^3-1, r = jp2q^2-1, s = jp2r^2-1, and jp2*s^2-1 are prime numbers.

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A255258 Expansion of q^2 * phi(q) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.

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cp's OEIS Frontend

A224610 Smallest j such that j*2*prime(n)^3-1 and j*2*prime(n)*q^2-1 are prime.

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A224611 Smallest j such that j*2*p(n)^3-1=q is prime, j*2*p(n)*q^2-1=r, j*2*p(n)*r^2-1=s, where r and s are also prime.

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A224612 Let p = prime(n). Smallest j such that q = j*2*p^3-1, r = j*p*2*q^2-1, s = j*p*2*r^2-1, and j*p*2*s^2-1 are prime numbers.

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Extensions

A255258 Expansion of q^2 * phi(q) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.

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A224610 Smallest j such that j2prime(n)^3-1 and j2prime(n)*q^2-1 are prime.

A224611 Smallest j such that j2p(n)^3-1=q is prime, j2p(n)q^2-1=r, j2p(n)r^2-1=s, where r and s are also prime.

A224612 Let p = prime(n). Smallest j such that q = j2p^3-1, r = jp2q^2-1, s = jp2r^2-1, and jp2*s^2-1 are prime numbers.