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

%I #7 Sep 15 2024 02:31:01

%S 64,192,243,320,441,448,486,704,832,882,960,1088,1215,1216,1344,1472,

%T 1701,1764,1856,1984,2112,2205,2240,2368,2430,2496,2624,2673,2752,

%U 3008,3159,3264,3392,3402,3520,3648,3776,3904,4131,4160,4288,4410,4416,4544,4617,4672,4851,4928

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

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

%C 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.

%C 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)).

%H Amiram Eldar, <a href="/A376204/b376204.txt">Table of n, a(n) for n = 1..10000</a>

%t 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[#]] &]

%o (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]);}

%o is(k) = {my(s1 = s(k)); s1 > 1 && ispowerful(s1);}

%Y Subsequence of A013929.

%Y A349109 \ {1} is a subsequence.

%Y Cf. A005117, A059956, A112526, A183097.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Sep 15 2024