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
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
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
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.