cp's OEIS Frontend

A348978 Numerator of ratio A332993(n) / sigma(n).

%I A348978 #14 Feb 15 2023 13:58:36
%S A348978 1,1,1,1,1,5,1,1,1,8,1,11,1,11,7,1,1,31,1,6,29,17,1,23,1,20,1,25,1,17,
%T A348978 1,1,15,26,43,67,1,29,53,38,1,71,1,13,11,35,1,47,1,27,23,46,1,47,67,
%U A348978 53,77,44,1,37,1,47,23,1,79,37,1,20,31,113,1,139,1,56,53,67,89,131,1,26,1,62,1,155,103,65,39,83
%N A348978 Numerator of ratio A332993(n) / sigma(n).
%C A348978 Ratio A332993(n) / sigma(n) tells how large proportion of the divisor sum we obtain if we sum just those divisors of n that can be obtained by repeatedly taking the largest proper divisor (of previous such divisor, starting from n, which is included in the sum), up to and including the terminal 1. Pair a(n) / A348979(n) shows the ratio in the lowest terms: 1/1, 1/1, 1/1, 1/1, 1/1, 5/6, 1/1, 1/1, 1/1, 8/9, 1/1, 11/14, 1/1, 11/12, 7/8, 1/1, 1/1, 31/39, 1/1, 6/7, 29/32, 17/18, 1/1, 23/30, etc. The ratio is 1 for all powers of primes (A000961).
%H A348978 Antti Karttunen, <a href="/A348978/b348978.txt">Table of n, a(n) for n = 1..16384</a>
%H A348978 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348978 a(n) = A332993(n) / A348977(n) = A332993(n) / gcd(A000203(n), A332993(n)).
%t A348978 f[n_] := n/FactorInteger[n][[1, 1]]; g[1] = 1; g[n_] := g[n] = n + g[f[n]]; a[n_] := Numerator[g[n]/DivisorSigma[1, n]]; Array[a, 100] (* _Amiram Eldar_, Nov 06 2021 *)
%o A348978 (PARI)
%o A348978 A332993(n) = if(1==n,n,n + A332993(n/vecmin(factor(n)[,1])));
%o A348978 A348978(n) = { my(u=A332993(n)); (u/gcd(sigma(n), u)); };
%Y A348978 Cf. A000203, A000961, A332993, A333783, A348977, A348979 (denominators).
%Y A348978 Cf. also A348988, A348989.
%K A348978 nonn,frac
%O A348978 1,6
%A A348978 _Antti Karttunen_, Nov 06 2021