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

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, 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, 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

Offset

1,6

Comments

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).

Links

Table of n, a(n) for n=1..88.

Index entries for sequences related to sigma(n)

Formula

a(n) = A332994(n) / A348987(n) = A332994(n) / gcd(A000203(n), A332994(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)
A332994(n) = if(1==n, n, n + A332994(n/vecmax(factor(n)[, 1])));
A348988(n) = { my(u=A332994(n)); (u/gcd(sigma(n), u)); };

Crossrefs

Cf. A000203, A000961, A332994, A333784, A348987, A348989 (denominators).

Cf. also A348978.

Sequence in context: A338817 A121585 A261959 * A257565 A276121 A262809

Adjacent sequences: A348985 A348986 A348987 * A348989 A348990 A348991

Keyword

nonn,frac

Author

Antti Karttunen, Nov 06 2021

Status

approved