login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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.
2

%I #14 Jul 29 2024 06:14:44

%S 1,2,8,9,10,18,24,27,30,54,165,238,288,512,656,660,864,952,1536,1968,

%T 2464,2880,4608,4680,13824,14448,14976,16728,19008,19992,23040,29376,

%U 60928,152064,155520,172368,279552,474936,746928,1070592,1114560,1524096,1703520

%N 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 Amiram Eldar, <a href="/A369204/b369204.txt">Table of n, a(n) for n = 1..80</a>

%e A188999(18) = 4 * 10 = 40 and A034448(40) = 9 * 6 = 54 = 3 * 18, so 18 is a term with k = 3.

%o (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 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 isok(n) = (a034448(a188999(n))%n) == 0;

%Y Cf. A038843 (analog for A034448(A034448(m))), A318175 (analog for A188999(A188999(m))).

%Y Cf. A369205 (analog for A188999(A034448(m))).

%K nonn

%O 1,2

%A _Tomohiro Yamada_, Jan 16 2024