A379833 The second Jordan totient function applied to the squares.

1, 12, 72, 192, 600, 864, 2352, 3072, 5832, 7200, 14520, 13824, 28392, 28224, 43200, 49152, 83232, 69984, 129960, 115200, 169344, 174240, 279312, 221184, 375000, 340704, 472392, 451584, 706440, 518400, 922560, 786432, 1045440, 998784, 1411200, 1119744, 1872792

Offset: 1

Author

Amiram Eldar, Jan 03 2025

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A002117, A002618 (analogous with J_1 = phi), A007434, A379715, A379716, A379717, A379718, A379832.

Programs

Maple

a:= n-> mul((i[1]^2-1)*i[1]^(4*i[2]-2), i=ifactors(n)[2]):
seq(a(n), n=1..37);  # Alois P. Heinz, Jan 03 2025

Mathematica

f[p_, e_] := (p^2 - 1) * p^(4*e - 2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(4*f[i,2] - 2));}

Formula

a(n) = J_2(n^2) = A007434(A000290(n)).

Multiplicative with a(p^e) = (p^2-1) * p^(4*e-2).

Dirichlet g.f.: zeta(s-4)/zeta(s-2).

In general, Dirichlet g.f. of J_k(n^m): zeta(s-m*k)/zeta(s-m*k+k), where J_k is the k-th Jordan totient function.

Sum_{i=1..n} a(i) ~ n^5 / (5*zeta(3)).

In general, Sum_{i=1..n} J_k(i^m) ~ n^(k*m+1) / ((k*m+1)*zeta(k+1)) for k,m >= 1.

Sum_{n>=1} 1/a(n) = (Pi^6/540) * Product_{p prime} (1 - 1/p^2 + 1/p^6) = 1.10666099915727116962...

In general, Sum_{n>=1} 1/J_k(n^m) = zeta(k) * zeta(k*m) * Product_{p prime} (1 - 1/p^k + 1/p^(k*m+k)), for k,m >= 2, and zeta(2) * zeta(m) * Product_{p prime} (1 - 1/p^2 + 1/p^(m+1) + 1/p^(m+2)) for k = 1 and m >= 2.