cp's OEIS Frontend

A341526 Numerator of ratio n*sigma(A003961(n)) / sigma(n)*A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.

%I A341526 #15 Feb 17 2021 20:28:43
%S A341526 1,8,9,52,20,4,21,64,279,160,77,26,117,28,6,1936,170,248,114,1040,189,
%T A341526 308,115,32,1425,104,1053,26,464,16,589,1664,231,1360,10,124,777,304,
%U A341526 1053,1280,902,42,516,22,372,230,423,968,343,3800,17,676,530,468,110,224,513,3712,1829,104,2074,589,5859,69952,780,154
%N A341526 Numerator of ratio n*sigma(A003961(n)) / sigma(n)*A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.
%C A341526 Like the ratios sigma(n)/n, A003973(n)/A003961(n) and A003961(n)/n, also the ratio r(n) = A341528(n)/A341529(n) is multiplicative: if gcd(x,y) = 1, r(x*y) = r(x)*r(y).
%H A341526 Antti Karttunen, <a href="/A341526/b341526.txt">Table of n, a(n) for n = 1..8191</a>
%H A341526 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A341526 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A341526 a(n) = A341528(n) / A341530(n) = A341528(n) / gcd(A341528(n), A341529(n)).
%F A341526 For all n > 1, a(n) < A341527(n).
%t A341526 f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator[n*DivisorSigma[1, (gn = g[n])]/(DivisorSigma[1, n] * gn)]; Array[a, 100] (* _Amiram Eldar_, Feb 17 2021 *)
%o A341526 (PARI)
%o A341526 A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A341526 A341526(n) = { my(s=A003961(n)); numerator((sigma(s)*n)/(sigma(n)*s)); };
%Y A341526 Cf. A000203, A003961, A003973, A017665, A017666, A336849, A341525, A341528, A341529, A341530.
%Y A341526 Cf. A341527 (denominators).
%Y A341526 Cf. A341626 (same sequence as applied onto prime shift array A246278).
%K A341526 nonn,frac
%O A341526 1,2
%A A341526 _Antti Karttunen_, Feb 16 2021