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A326042 a(n) = A064989(sigma(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.

%I A326042 #67 Dec 15 2024 17:15:27
%S A326042 1,1,2,11,1,2,2,3,29,1,5,22,4,2,2,49,3,29,2,11,4,5,6,6,34,4,22,22,1,2,
%T A326042 17,55,10,3,2,319,10,2,8,3,7,4,2,55,29,6,8,98,85,34,6,44,6,22,5,6,4,1,
%U A326042 29,22,13,17,58,1091,4,10,4,33,12,2,31,87,3,10,68,22,10,8,10,49,469,7,12,44,3,2,2,15,25,29,8,66,34,8
%N A326042 a(n) = A064989(sigma(A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, and A064989 shifts it back towards smaller primes.
%C A326042 For any other number n than those in A326182 we have a(n) < A003961(n).
%C A326042 Fixed points k (for which a(k) = k) satisfy A003973(k) = 2^e * A003961(k) for some exponent e >= 0. Applying A003961 to such numbers gives the odd terms in A336702, of which there are likely to be just a single instance, its initial 1. (Clarified Nov 07 2021).
%C A326042 Conjecture: There are no other fixed points than a(1) = 1. If true, then there are no odd perfect numbers. This condition is equivalent to the condition that if A161942 has no fixed points larger than one, then there are no odd perfect numbers. This follows as whenever k is a fixed point, that is, a(k) = k, then we should also have A003961(a(k)) = A003961(A064989(sigma(A003961(k)))) = A161942(A003961(k)) = A003961(k). Note that A003961 is an injective and surjective mapping from natural numbers to odd numbers, A064989 is its (left) inverse, and composition A003961(A064989(n)) is equivalent to A000265(n).
%C A326042 From _Antti Karttunen_, Aug 05 2020: (Start)
%C A326042 For any hypothetical odd perfect number x, we would have A003973(k) = 2 * A003961(k), with k = A064989(x) and x = A003961(k). Thus we would have a(k) = A064989(sigma(A003961(k))) = A064989(sigma(x)) = A064989(2*x) = A064989(x) = k. On the other hand, A003973(k) = sigma(A003961(k)) < A003961(A003961(k)) [see A286385 for the reason why], so a necessary condition for this is that x should be one of the terms of A246282. (Clarified Dec 01 2020).
%C A326042 (End)
%H A326042 Antti Karttunen, <a href="/A326042/b326042.txt">Table of n, a(n) for n = 1..20000</a>
%H A326042 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A326042 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A326042 a(n) = A064989(A003973(n)) = A064989(sigma(A003961(n))).
%F A326042 For k in A000037, a(k) = A064989(A003973(k)/2) = A064989((1/2)*sigma(A003961(k))).
%F A326042 Multiplicative with a(p^e) = A064989((q^(e+1)-1)/(q-1)), where q = nextPrime(p). - _Antti Karttunen_, Nov 05 2021
%F A326042 a(n) = A353790(n) / A353767(n) = A353794(n) / A351456(n). - _Antti Karttunen_, May 13 2022
%t A326042 f1[p_, e_] := NextPrime[p]^e; a1[1] = 1; a1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := a2[DivisorSigma[1, a1[n]]]; Array[a, 100] (* _Amiram Eldar_, Nov 07 2021 *)
%o A326042 (PARI)
%o A326042 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A326042 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A326042 A326042(n) = A064989(sigma(A003961(n)));
%Y A326042 Cf. A000037, A000203, A000265, A000593, A003961, A003973, A064989, A161942, A162284, A246282, A286385, A326041, A326182, A336702 (numbers whose abundancy index is a power of 2).
%Y A326042 Cf. A348736 [n - a(n)], A348738 [a(n) < n], A348739 [a(n) > n], A348750 [= A064989(a(A003961(n)))], A348940 [gcd(n,a(n))], A348941, A348942, A351456, A353767, A353790, A353794.
%Y A326042 Cf. also A332223 for another conjugation of sigma.
%K A326042 nonn,mult
%O A326042 1,3
%A A326042 _Antti Karttunen_, Jun 16 2019
%E A326042 Keyword:mult added by _Antti Karttunen_, Nov 05 2021