A371412 -id:A371412 - cp's OEIS frontend

A379718 The second Jordan totient function applied to the cubefull numbers: a(n) = A007434(A036966(n)).

1, 48, 192, 648, 768, 3072, 5832, 15000, 12288, 31104, 52488, 49152, 115248, 124416, 196608, 375000, 279936, 472392, 497664, 720000, 786432, 1119744, 1756920, 1990656, 2519424, 2880000, 3145728, 4251528, 4798248, 5647152, 4478976, 5531904, 9375000, 9720000, 7962624

Offset: 1

Amiram Eldar, Dec 31 2024

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

Cf. A007434, A013661, A036966, A371412 (analogous with J_1 = phi), A379715, A379716, A379717.

f[p_, e_] := (p^2-1) * p^(2*e-2); j2[1] = 1; j2[n_] := Times @@ f @@@ FactorInteger[n]; Join[{1}, j2 /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]

j2(f) = prod(i = 1, #f~, (f[i,1]^2 - 1) * f[i,1]^(2*f[i,2] - 2));
list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmin(f[, 2]) > 2, print1(j2(f), ", "))); }

Sum_{n>=1} 1/a(n) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^6) = 1.02964361441212748276... .
In general, Sum_{m cubefull} 1/J_k(m) = zeta(k)^2 * Product_{p prime} (1 - 2/p^k + 1/p^(2*k) + 1/p^(3*k)), for k >= 2, where J_k is the k-th Jordan totient function.
In general, Sum_{m k-full} 1/J_2(m) = zeta(2)^2 * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^(2*k)), for k >= 2.

A371413 Dedekind psi function applied to the cubefull numbers (A036966).

Original entry on oeis.org

1, 12, 24, 36, 48, 96, 108, 150, 192, 432, 324, 384, 392, 864, 768, 750, 1296, 972, 1728, 1800, 1536, 2592, 1452, 3456, 3888, 3600, 3072, 2916, 2366, 2744, 5184, 4704, 3750, 5400, 6912, 7776, 7200, 6144, 5202, 9000, 10368, 9408, 11664, 8748, 7220, 13824, 15552

Offset: 1

Amiram Eldar, Mar 22 2024

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

Cf. A001615, A036966, A065464.

Similar sequences: A323332, A371412, A371415.

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Join[{1}, psi /@ Select[Range[20000], AllTrue[Last /@ FactorInteger[#], #1 > 2 &] &]]
(* or *)
f[n_] := Module[{f = FactorInteger[n], p, e}, If[n == 1, 1, p = f[[;;, 1]]; e = f[[;;, 2]]; If[Min[e] > 2, Times @@ ((p+1) * p^(e-1)), Nothing]]]; Array[f, 20000]

dedpsi(f) = prod(i = 1, #f~, (f[i, 1] + 1) * f[i, 1]^(f[i, 2]-1));
lista(max) = {my(f); print1(1, ", "); for(k = 2, max, f = factor(k); if(vecmin(f[, 2]) > 2, print1(dedpsi(f), ", "))); }

a(n) = A001615(A036966(n)).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/((p^2-1)*p)) = 1.231291... (A065487).

cp's OEIS Frontend

A379718 The second Jordan totient function applied to the cubefull numbers: a(n) = A007434(A036966(n)).

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