A376204 Numbers whose sum of powerful divisors (including 1) is a powerful number that is larger than 1.

64, 192, 243, 320, 441, 448, 486, 704, 832, 882, 960, 1088, 1215, 1216, 1344, 1472, 1701, 1764, 1856, 1984, 2112, 2205, 2240, 2368, 2430, 2496, 2624, 2673, 2752, 3008, 3159, 3264, 3392, 3402, 3520, 3648, 3776, 3904, 4131, 4160, 4288, 4410, 4416, 4544, 4617, 4672, 4851, 4928

Offset: 1

Author

Amiram Eldar, Sep 15 2024

Keywords

• nonn
• easy

Comments

Numbers k such that A112526(A183097(k)) = 1.

The primitive terms of this sequence are the powerful terms (A349109 \ {1}). If m > 1 is a powerful term then k*m is a term of this sequence for all squarefree numbers k that are coprime to m.

The asymptotic density of this sequence is Sum_{i>=2} f(A349109(i))/A349109(i) = 0.00935344863979..., where f(k) = (6/Pi^2) * Product_{p|k} (p/(p+1)).

Links

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

Crossrefs

Subsequence of A013929.

A349109 \ {1} is a subsequence.

Cf. A005117, A059956, A112526, A183097.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[k_] := Times @@ f @@@ FactorInteger[k]; q[k_] := AllTrue[FactorInteger[k][[;; , 2]], # > 1 &]; Select[Range[5000], q[s[#]] &]

PARI

s(k) = {my(f = factor(k)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1) - f[i,1]);}
is(k) = {my(s1 = s(k)); s1 > 1 && ispowerful(s1);}