A348978 - OEIS

%I #14 Feb 15 2023 13:58:36

%S 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 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 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 Numerator of ratio A332993(n) / sigma(n).

%C 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 Antti Karttunen, <a href="/A348978/b348978.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A332993(n) / A348977(n) = A332993(n) / gcd(A000203(n), A332993(n)).

%t 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 (PARI)

%o A332993(n) = if(1==n,n,n + A332993(n/vecmin(factor(n)[,1])));

%o A348978(n) = { my(u=A332993(n)); (u/gcd(sigma(n), u)); };

%Y Cf. A000203, A000961, A332993, A333783, A348977, A348979 (denominators).

%Y Cf. also A348988, A348989.

%K nonn,frac

%O 1,6

%A _Antti Karttunen_, Nov 06 2021