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A376204
Numbers whose sum of powerful divisors (including 1) is a powerful number that is larger than 1.
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
1,1
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
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[#]] &]
PROG
(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); }
CROSSREFS
Subsequence of A013929.
A349109 \ {1} is a subsequence.
Sequence in context: A202332 A208636 A135270 * A045052 A195089 A336596
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 15 2024
STATUS
approved