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

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, 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, 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

Offset

1,6

Comments

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences related to sigma(n)

Formula

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

Mathematica

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 *)

Prog

(PARI)
A332993(n) = if(1==n, n, n + A332993(n/vecmin(factor(n)[, 1])));
A348978(n) = { my(u=A332993(n)); (u/gcd(sigma(n), u)); };

Crossrefs

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

Cf. also A348988, A348989.

Sequence in context: A292771 A369364 A322837 * A168677 A345939 A140210

Adjacent sequences: A348975 A348976 A348977 * A348979 A348980 A348981

Keyword

nonn,frac

Author

Antti Karttunen, Nov 06 2021

Status

approved