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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.

%I A354195 #13 May 25 2022 22:51:19
%S A354195 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,
%T A354195 25,20,113,6,25,5,58,20,5,2,20,15,226,10,220,29,10,6,12,6,170,3,15,12,
%U A354195 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
%N 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.
%C A354195 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.
%H A354195 Antti Karttunen, <a href="/A354195/b354195.txt">Table of n, a(n) for n = 1..20000</a>
%H A354195 Antti Karttunen, <a href="/A354195/a354195.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H A354195 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A354195 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A354195 a(n) = A064989(A051027(A003961(n))).
%o A354195 (PARI)
%o A354195 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A354195 A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
%o A354195 A354195(n) = A064989(sigma(sigma(A003961(n))));
%Y A354195 Cf. A000203, A003961, A051027, A064989, A354196 [= A064989(a(A003961(n)))], A354346 [= 2*n - a(n)].
%Y A354195 Cf. also A326042, A354197, A354199.
%K A354195 nonn
%O A354195 1,2
%A A354195 _Antti Karttunen_, May 23 2022