A393084 Primes p such that c(p) - tau(p) is divisible by 120, where c(n) are coefficients of Klein's j-invariant and tau(n) are values of the Ramanujan tau function.

479, 18839, 43391, 67679, 76871, 94079, 117071, 135479, 139991, 151871, 154991, 159791, 189311, 201359, 248639, 253679, 308999, 336599, 379319, 405599, 418751, 432359, 435191, 463031, 485831, 506999, 507599, 537599, 558599, 593951, 607991, 627911, 634079, 659831, 680759

Offset

1,1

Comments

Primes p such that A000521(p) - A000594(p) is divisible by 120.

Let j(q)=q^(-1)+744+Sum_{n>=1} c(n) q^n and Delta(q)=Sum_{n>=1} tau(n) q^n = q*Prod_{m>=1} (1-q^m)^24.

Implementation note: to test membership it suffices to compute c(p) (mod 720) and tau(p) (mod 720), then check (c(p)-tau(p)) (mod 120). One may use j(q)=E4(q)^3/Delta(q) and E4(q)=1+240*(...) to get j(q) congruent to 1/Delta(q) (mod 720).

Links

Daniyar Supiyev, Table of n, a(n) for n = 1..3506

Wikipedia, Klein modular invariant (j-invariant).

Wikipedia, Ramanujan tau function.

Example

p=479 is in the sequence because (c(p)-tau(p)) mod 720 = 240, hence c(p)-tau(p) is congruent to 0 (mod 120) but not congruent to 0 (mod 720).
The first prime with c(p)-tau(p) congruent to 0 (mod 720) is p=18839.

Prog

(PARI)
\\ phase primes up to N: primes p with (c(p)-tau(p)) == 0 (mod step),
\\ where c(n) is from j(q) and tau(n) from Delta(q).
\\ We compute everything mod 720 via:
\\   Delta(q) = q * prod_{n>=1} (1-q^n)^24 = q * pow24(q)
\\   1/Delta(q) = q^-1 * invpow(q), where invpow(q) = 1/pow24(q)
\\ Justification: j(q)=E4(q)^3/Delta(q) and E4(q)=1+240*(...) => E4(q)^3 == 1 (mod 720), so j(q) == 1/Delta(q) (mod 720).
\\ Then (mod 720) we can take c(p) as coeff_{q^p} (1/Delta), i.e. polcoeff(invpow, p+1),
\\ and tau(p) as coeff_{q^p} Delta, i.e. polcoeff(pow24, p-1).
phase_primes_upto(N, step)=
{
  my(m=720, x='x, K, g, pow24, invpow, out=List(), k, s, p1, p2, p, tau, c);
  K = floor((sqrt(24*(N+1)+1)+1)/6)+2; \\ enough for pentagonal exponents <= N+1
  \\ g(q)=prod_{n>=1}(1-q^n) via Euler pentagonal theorem
  g = 1 + O(x^(N+2));
  for(k=1, K,
    s = if(k%2, -1, 1);
    p1 = k*(3*k-1)/2;
    p2 = k*(3*k+1)/2;
    if(p1<=N+1, g += s*x^p1);
    if(p2<=N+1, g += s*x^p2)
  );
  g = Mod(g, m);
  pow24 = g^24;      \\ prod (1-q^n)^24  (this is Delta(q)/q)
  invpow = 1/pow24;  \\ prod (1-q^n)^(-24) (this is q/Delta(q))
  forprime(p=5, N,
    tau = polcoeff(pow24, p-1);
    c   = polcoeff(invpow, p+1);
    if(Mod(c - tau, m) % step == 0, listput(out, p))
  );
  Vec(out);
}

Crossrefs

Cf. A000040, A000521, A000594.

Sequence in context: A395529 A377476 A108256 * A262222 A115157 A019287

Adjacent sequences: A393081 A393082 A393083 * A393085 A393086 A393087

Keyword

nonn

Author

Daniyar Supiyev, Mar 01 2026

Status

approved