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a(n) = A064989(A064989(sigma(sigma(sigma(A003961(A003961(n))))))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.
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%I #11 May 24 2022 16:31:48

%S 1,3,1,3,3,3,2,26,23,3,3,3,1,3,21,6,3,9,14,22,2,2,7,182,3,14,313,201,

%T 3,3,3,603,3,3,3,115,3,3,2,3,3,21,2,9,9,3,2,75,2,22,3,109,3,21,46,109,

%U 2,23,7,154,3,6,22,222,2,14,2,22,29,6,1,78,3,161,69,1407,6,2,21,44,7,21,14,201,21,39,3,529

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

%C For any hypothetical odd perfect number opn that is not a multiple of 3, it holds that a(n) = A354196(n) = A348750(n) = n, where n = A064989(A064989(opn)). See also comments in A353365.

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A064989(A354197(A003961(n))) = A064989(A064989(A066971(A003961(A003961(n))))).

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };

%o A354198(n) = A064989(A064989(sigma(sigma(sigma(A003961(A003961(n)))))));

%Y Cf. A000203, A003961, A064989, A066971, A326042, A353365, A354197.

%Y Cf. also A348750, A354196.

%K nonn

%O 1,2

%A _Antti Karttunen_, May 24 2022