A349109 -id:A349109 - cp's OEIS frontend

A349110 Powerful numbers (A001694) whose sum of aliquot powerful divisors (including 1) is larger than 1 and is also powerful.

128, 729, 900, 4900, 10404, 17424, 24336, 52900, 78400, 79524, 81796, 297025, 304175, 304200, 313600, 346921, 417316, 532900, 1612900, 1656200, 1960000, 2238016, 2464900, 3129361, 3232804, 3334276, 3496900, 3534400, 3992004, 6056521, 6974881, 9245000, 10672200

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

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

			128 = 2^7 is a term since it is powerful and the sum of its aliquot powerful divisors, A183097(128) - 128 =  1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.

Cf. A001694, A183097, A349109.

powQ[n_] := AllTrue[FactorInteger[n][[;;,2]], # > 1 &]; f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := powQ[n] && powQ[s[n] - n]; Select[Range[1.1*10^7], q]

isok(n) = my(s); ispowerful(n) && (s=sumdiv(n, d, if (d1) && ispowerful(s); \\ Michel Marcus, Nov 08 2021

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

Original entry on oeis.org

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

Amiram Eldar, Sep 15 2024

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)).

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

Subsequence of A013929.
A349109 \ {1} is a subsequence.
Cf. A005117, A059956, A112526, A183097.

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[#]] &]

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);}

cp's OEIS Frontend

A349110 Powerful numbers (A001694) whose sum of aliquot powerful divisors (including 1) is larger than 1 and is also powerful.

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