cp's OEIS Frontend

A348988 Numerator of A332994(n) / sigma(n).

%I A348988 #13 Jan 24 2025 08:44:01
%S A348988 1,1,1,1,1,3,1,1,1,13,1,19,1,17,19,1,1,9,1,9,25,25,1,13,1,29,1,5,1,13,
%T A348988 1,1,37,37,41,55,1,41,43,11,1,17,1,17,29,49,1,79,1,21,55,59,1,27,61,
%U A348988 71,61,61,1,79,1,65,19,1,71,25,1,25,73,83,1,37,1,77,47,83,85,29,1,37,1,85,1,103,91,89,91,103
%N A348988 Numerator of A332994(n) / sigma(n).
%C A348988 Ratio A332994(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 dividing a single instance of the largest prime divisor out of previous such divisor (when starting from n, which is included in the sum), up to and including the terminal 1. Pair a(n) / A348989(n) shows the ratio in the lowest terms: 1/1, 1/1, 1/1, 1/1, 1/1, 3/4, 1/1, 1/1, 1/1, 13/18, 1/1, 19/28, 1/1, 17/24, 19/24, 1/1, 1/1, 9/13, 1/1, 9/14, 25/32, 25/36, 1/1, 13/20, 1/1, 29/42, 1/1, 5/8, 1/1, 13/24, 1/1, etc. The ratio is 1 for all powers of primes (A000961).
%H A348988 Antti Karttunen, <a href="/A348988/b348988.txt">Table of n, a(n) for n = 1..20000</a>
%H A348988 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348988 a(n) = A332994(n) / A348987(n) = A332994(n) / gcd(A000203(n), A332994(n)).
%t A348988 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 A348988 (PARI)
%o A348988 A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));
%o A348988 A348988(n) = { my(u=A332994(n)); (u/gcd(sigma(n), u)); };
%Y A348988 Cf. A000203, A000961, A332994, A333784, A348987, A348989 (denominators).
%Y A348988 Cf. also A348978.
%K A348988 nonn,frac
%O A348988 1,6
%A A348988 _Antti Karttunen_, Nov 06 2021