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A369204 Numbers m such that A034448(A188999(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

%I A369204 #14 Jul 29 2024 06:14:44
%S A369204 1,2,8,9,10,18,24,27,30,54,165,238,288,512,656,660,864,952,1536,1968,
%T A369204 2464,2880,4608,4680,13824,14448,14976,16728,19008,19992,23040,29376,
%U A369204 60928,152064,155520,172368,279552,474936,746928,1070592,1114560,1524096,1703520
%N A369204 Numbers m such that A034448(A188999(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.
%H A369204 Amiram Eldar, <a href="/A369204/b369204.txt">Table of n, a(n) for n = 1..80</a>
%e A369204 A188999(18) = 4 * 10 = 40 and A034448(40) = 9 * 6 = 54 = 3 * 18, so 18 is a term with k = 3.
%o A369204 (PARI) a034448(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=p^e+1;f[i,2]=1);factorback(f)};
%o A369204 a188999(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=if(e%2,(p^(e+1)-1)/(p-1),(p^(e+1)-1)/(p-1)-p^(e/2));f[i,2]=1);factorback(f)};
%o A369204 isok(n) = (a034448(a188999(n))%n) == 0;
%Y A369204 Cf. A038843 (analog for A034448(A034448(m))), A318175 (analog for A188999(A188999(m))).
%Y A369204 Cf. A369205 (analog for A188999(A034448(m))).
%K A369204 nonn
%O A369204 1,2
%A A369204 _Tomohiro Yamada_, Jan 16 2024