OFFSET
1,2
COMMENTS
Numbers n such that A324054(n-1) is equal to A324108(n), which is a multiplicative function with A324108(p^e) = A324054((p^e)-1).
Prime powers (A000961) is a subsequence by definition.
Also A070776 is a subsequence. This follows because for every n of the form 2^i * p^j (where p is an odd prime, and i >= 0, j >= 0), we have A324108(2^i * p^j) = A324054(2^i - 1)*A324054(p^j - 1) = sigma(A005940(2^i)) * sigma(A005940(p^j)). Because A005940(1) = 1, and A005940(2n) = 2*A005940(n), the powers of two are among the fixed points of A005940 (cf. A029747), thus the left half of product is sigma(2^i), while on the other hand, we know that A005940(p^j) is odd (because A005940 also preserves parity), and thus the whole product is equal to sigma(2^i * A005940(p^j)) = sigma(A005940(2^i * p^j)) = A324054((2^i * p^j)-1).
See subsequence A324111 for less regular solutions.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10001
PROG
(PARI)
A324054(n) = { my(p=2, mp=p*p, m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4), mp *= p, m *= (mp-1)/(p-1))); n>>=1); (m); };
for(n=1, 121, if(isA324109(n), print1(n, ", ")));
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen and David A. Corneth, Feb 15 2019
STATUS
approved