A337044 -id:A337044 - cp's OEIS frontend

A337045 Indecomposable sigma-powerful numbers: powerful numbers k such that sigma(k) is also powerful, but restricted to terms that are not the product of 2 terms > 1 of A337044.

81, 343, 400, 9261, 189728, 224939, 972000, 1705636, 2205472, 3087000, 3591200, 3648100, 7968032, 13645088, 15350724, 21161304, 24240600, 25992000, 26680500, 29184800, 32832900, 48586824, 51595489, 80802000, 103617387, 109215352, 110215125, 119604096, 122805792

Offset: 1

Hugo Pfoertner, Aug 15 2020

nonn

This is an implementation of the suggestion that Walter A. Kehowski made on his website (see link) with regard to so-called indecomposable sigma-powerful numbers. However, the results deviate from the table linked there. The table is considered to be deficient.

			From _David A. Corneth_, Aug 29 2020: (Start)
No two proper divisors of 400 are sigma-powerful and have the product of those divisors 400 so 400 is in the sequence.
27783 = 81 * 343 is sigma-powerful but 81 and 343 are sigma-powerful as well so 27783 can be decomposed into two sigma-powerful factors. So 27783 is not in the sequence. (End)

David A. Corneth, Table of n, a(n) for n = 1..6421 (terms < 10^18)
Walter A. Kehowski, Sigma-powerful numbers, Aug 09 2010.

Cf. A000203, A001694, A128607, A180090, A337044.

v=vector(50); n=0;
for(m=2, 150000000, my(is); if(ispowerful(m) && ispowerful(sigma(m)), v[n++]=m; is=1; for(j=1, n-1, if(v[n]%v[j], , if(vecsearch(v[1..n-1], v[n]/v[j]), is=0; break))); if(is, print1(v[n], ", "))))

A349109 Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.

Original entry on oeis.org

1, 64, 243, 441, 1764, 9800, 15552, 28224, 41616, 60516, 82369, 88200, 189728, 226576, 329476, 336200, 648675, 741321, 968256, 1317904, 1428025, 1707552, 1943236, 2039184, 2056356, 2381400, 2446227, 2798929, 2965284, 2986568, 4372281, 5189400, 5271616, 6508832

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

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

			64 = 2^6 is a term since it is powerful and the sum of its powerful divisors, A183097(64) =  1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.

Amiram Eldar, Table of n, a(n) for n = 1..12154 (terms below 10^19)

Cf. A001694, A112526, A180090, A183097, A337044, A337045, A349110.

powQ[n_] := n == 1 || 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]]; Select[Range[7*10^6], q]

isok(n) = ispowerful(n) && ispowerful(sumdiv(n, d, d*ispowerful(d))); \\ Michel Marcus, Nov 08 2021

is(k) = {my(f = factor(k)); ispowerful(f) && ispowerful(prod(i = 1, #f~, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1) - f[i,1]));} \\ Amiram Eldar, Sep 14 2024

cp's OEIS Frontend

A337045 Indecomposable sigma-powerful numbers: powerful numbers k such that sigma(k) is also powerful, but restricted to terms that are not the product of 2 terms > 1 of A337044.

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A349109 Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI

PARI