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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.

%I A337045 #10 Aug 31 2020 12:30:24
%S A337045 81,343,400,9261,189728,224939,972000,1705636,2205472,3087000,3591200,
%T A337045 3648100,7968032,13645088,15350724,21161304,24240600,25992000,
%U A337045 26680500,29184800,32832900,48586824,51595489,80802000,103617387,109215352,110215125,119604096,122805792
%N 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.
%C A337045 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.
%H A337045 David A. Corneth, <a href="/A337045/b337045.txt">Table of n, a(n) for n = 1..6421</a> (terms < 10^18)
%H A337045 Walter A. Kehowski, <a href="http://web.gccaz.edu/~walof42631/Sigma-powerful-numbers.html">Sigma-powerful numbers</a>, Aug 09 2010.
%e A337045 From _David A. Corneth_, Aug 29 2020: (Start)
%e A337045 No two proper divisors of 400 are sigma-powerful and have the product of those divisors 400 so 400 is in the sequence.
%e A337045 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)
%o A337045 (PARI) v=vector(50); n=0;
%o A337045 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], ", "))))
%Y A337045 Cf. A000203, A001694, A128607, A180090, A337044.
%K A337045 nonn
%O A337045 1,1
%A A337045 _Hugo Pfoertner_, Aug 15 2020