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A349161 a(n) = A003961(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.

%I A349161 #25 Mar 18 2023 03:26:13
%S A349161 1,1,5,9,7,5,11,9,25,7,13,45,17,11,35,81,19,25,23,3,55,13,29,9,49,17,
%T A349161 25,99,31,35,37,27,65,19,77,225,41,23,85,21,43,55,47,39,175,29,53,405,
%U A349161 121,49,95,153,59,25,91,99,23,31,61,15,67,37,275,729,17,65,71,19,145,77,73,45,79,41,245,207,143,85,83
%N A349161 a(n) = A003961(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 A349161 Numerator of ratio A003961(n) / A000203(n). Sequence A349162 gives the denominators.
%C A349161 Numerator of ratio A003961(n) / A161942(n). Sequence A348992 gives the denominators.
%C A349161 Both ratios are multiplicative because the constituent sequences are.
%C A349161 No 1's occur as terms after a(2), because for n > 2, sigma(n) < A003961(n). (See A286385).
%H A349161 Antti Karttunen, <a href="/A349161/b349161.txt">Table of n, a(n) for n = 1..20000</a>
%H A349161 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349161 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A349161 a(n) = A003961(n) / A342671(n) = A003961(n) / gcd(A000203(n), A003961(n)).
%F A349161 a(n) = A003961(A349164(n)).
%t A349161 Array[#2/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 79] (* _Michael De Vlieger_, Nov 11 2021 *)
%o A349161 (PARI)
%o A349161 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A349161 A349161(n) = { my(u=A003961(n)); (u/gcd(u,sigma(n))); };
%o A349161 (Python)
%o A349161 from math import prod, gcd
%o A349161 from sympy import nextprime, factorint
%o A349161 def A349161(n):
%o A349161     f = factorint(n).items()
%o A349161     a = prod(nextprime(p)**e for p, e in f)
%o A349161     b = prod((p**(e+1)-1)//(p-1) for p, e in f)
%o A349161     return a//gcd(a,b) # _Chai Wah Wu_, Mar 17 2023
%Y A349161 Cf. A000203, A003961, A161942, A286385, A319626, A319627, A326042, A336702, A342671, A349163, A349164, A349627, A349628.
%Y A349161 Cf. A348992, A349162 (denominators).
%K A349161 nonn,frac
%O A349161 1,3
%A A349161 _Antti Karttunen_, Nov 09 2021