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A350072 a(n) = sigma(n^2) / gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

%I A350072 #29 Jul 15 2023 21:59:46
%S A350072 1,7,13,31,31,91,57,127,121,31,133,403,183,133,403,511,307,847,381,
%T A350072 961,741,931,553,1651,781,427,1093,589,871,403,993,2047,133,2149,1767,
%U A350072 3751,1407,889,2379,3937,1723,1729,1893,4123,3751,3871,2257,6643,2801,781,3991,1891,2863,7651,589,2413,4953,6097,3541,12493
%N A350072 a(n) = sigma(n^2) / gcd(sigma(n^2), A003961(n^2)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.
%C A350072 Conjecture: There are no 1's after the initial term. Remark: If there were some k = x^2 > 1, for which a(x) = 1, then sigma(k) would be a divisor of A003961(k). In other words, d = A350073(k) = A064989(sigma(k)) would be a divisor of k. Then, if that divisor were also a unitary divisor [with gcd(d,k/d) = 1], it would need to satisfy the equation sigma(k) = sigma(d) * sigma(k/d) = sigma(A064989(sigma(k))) * sigma(k/A064989(sigma(k))), because sigma is a multiplicative function. (Minor correction by _Antti Karttunen_, Jul 11 2023)
%C A350072 Note that if d = A064989(sigma(k)) were a unitary divisor of a square k, then sigma(k) would also be a square, the cases which are quite rare (see A008848 and A336547). Also compare to A349756. - _Antti Karttunen_, Jul 24 2022
%H A350072 Antti Karttunen, <a href="/A350072/b350072.txt">Table of n, a(n) for n = 1..20000</a>
%H A350072 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A350072 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A350072 a(n) = A349162(n^2).
%F A350072 a(n) = A065764(n) / A350071(n).
%t A350072 f1[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^(2*e); a[1] = 1; a[n_] := (s = Times @@ f1 @@@ (f = FactorInteger[n])) / GCD[s, Times @@ f2 @@@ f]; Array[a, 60] (* _Amiram Eldar_, Dec 12 2021 *)
%o A350072 (PARI)
%o A350072 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A350072 A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };
%o A350072 A350072(n) = A349162(n^2);
%Y A350072 Cf. A000203, A003961, A008848, A064989, A065764, A349162, A350071, A350073.
%Y A350072 See also A069070, A336547, A349756, A364131.
%K A350072 nonn
%O A350072 1,2
%A A350072 _Antti Karttunen_, Dec 12 2021