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a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.
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%I #27 Feb 18 2022 11:10:24

%S 1,1,4,7,6,4,8,5,13,6,12,28,14,8,24,31,18,13,20,2,32,12,24,4,31,14,8,

%T 56,30,24,32,7,48,18,48,91,38,20,56,10,42,32,44,28,78,24,48,124,57,31,

%U 72,98,54,8,72,40,16,30,60,8,62,32,104,127,12,48,68,14,96,48,72,13,74,38,124,140,96,56,80,62,121,42

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

%C Denominator of ratio A003961(n) / A000203(n).

%C Small values are rare, but are not limited to the beginning. For example in range 1 .. 2^25, a(n) = 4 at n = 3, 6, 24, 792, 2720, 122944, 31307472.

%C Question: Would it be possible to prove that a(n) > 1 for all n > 2?

%C Obviously, 1's may occur only on squares & twice squares (A028982). See also comments in A350072. - _Antti Karttunen_, Feb 16 2022

%H Antti Karttunen, <a href="/A349162/b349162.txt">Table of n, a(n) for n = 1..22968</a>

%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) = A000203(n) / A342671(n) = A000203(n) / gcd(A000203(n), A003961(n)).

%t Array[#1/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 82] (* _Michael De Vlieger_, Nov 11 2021 *)

%o (PARI)

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

%o A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };

%Y Cf. A000203, A003961, A028982 (positions of odd terms), A319630, A336702, A342671, A348992 (the odd part), A348993, A349161 (numerators), A349163, A349164, A349627, A349628, A350072 [= a(n^2)].

%Y Cf. also A349745, A351551, A351554.

%K nonn,frac

%O 1,3

%A _Antti Karttunen_, Nov 09 2021