A354195 a(n) = A064989(sigma(sigma(A003961(n)))), where A003961 shifts the prime factorization one step towards larger primes, and A064989 shifts it back towards smaller primes.

1, 5, 2, 5, 6, 6, 5, 12, 1, 20, 2, 10, 22, 29, 29, 85, 10, 5, 6, 30, 66, 6, 4, 58, 3, 66, 25, 25, 20, 113, 6, 25, 5, 58, 20, 5, 2, 20, 15, 226, 10, 220, 29, 10, 6, 12, 6, 170, 3, 15, 12, 110, 10, 145, 29, 40, 319, 78, 2, 145, 20, 18, 5, 541, 319, 29, 66, 50, 110, 78, 34, 12, 58, 6, 66, 30, 6, 87, 5, 510, 8, 58, 44

Offset: 1

Author

Antti Karttunen, May 23 2022

Keywords

• nonn

Comments

For any hypothetical odd perfect number opn that is not a multiple of 3, it holds that a(n) = A354197(n) = 2*n, where n = A064989(opn) is an odd number.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A051027, A064989, A354196 [= A064989(a(A003961(n)))], A354346 [= 2*n - a(n)].

Cf. also A326042, A354197, A354199.

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A354195(n) = A064989(sigma(sigma(A003961(n))));

Formula

a(n) = A064989(A051027(A003961(n))).