A354198 -id:A354198 - cp's OEIS frontend

A354197 a(n) = A064989(sigma(sigma(sigma(A003961(n))))), where A003961 shifts the prime factorization one step towards larger primes, and A064989 shifts it back towards smaller primes.

1, 1, 5, 2, 2, 10, 5, 44, 20, 11, 6, 6, 5, 5, 5, 3, 2, 20, 10, 4, 10, 12, 66, 6, 58, 10, 204, 204, 11, 5, 10, 986, 20, 2, 55, 113, 20, 55, 12, 2, 5, 55, 5, 29, 40, 132, 12, 15, 40, 58, 132, 10, 6, 6, 6, 18, 5, 8, 20, 6, 22, 145, 78, 262, 5, 20, 10, 170, 10, 40, 6, 2486, 2, 40, 50, 12, 40, 12, 20, 6, 60, 5, 110, 20

Offset: 1

Antti Karttunen, May 24 2022

nonn

For any hypothetical odd perfect number opn that is not a multiple of 3, it holds that a(n) = A354195(n) = 2*n, where n = A064989(opn).

Cf. A000203, A003961, A064989, A066971, A354198 [= A064989(a(A003961(n)))].

Cf. also A326042, A354195.

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); };
A354197(n) = A064989(sigma(sigma(sigma(A003961(n)))));

a(n) = A064989(A066971(A003961(n))).

cp's OEIS Frontend

A354197 a(n) = A064989(sigma(sigma(sigma(A003961(n))))), where A003961 shifts the prime factorization one step towards larger primes, and A064989 shifts it back towards smaller primes.

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