A348987 -id:A348987 - cp's OEIS frontend

A348988 Numerator of 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

Antti Karttunen, Nov 06 2021

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).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

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

Cf. also 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 *)

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

a(n) = A332994(n) / A348987(n) = A332994(n) / gcd(A000203(n), A332994(n)).

A348989 Denominator of A332994(n) / sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 18, 1, 28, 1, 24, 24, 1, 1, 13, 1, 14, 32, 36, 1, 20, 1, 42, 1, 8, 1, 24, 1, 1, 48, 54, 48, 91, 1, 60, 56, 18, 1, 32, 1, 28, 39, 72, 1, 124, 1, 31, 72, 98, 1, 40, 72, 120, 80, 90, 1, 168, 1, 96, 26, 1, 84, 48, 1, 42, 96, 144, 1, 65, 1, 114, 62, 140, 96, 56, 1, 62, 1, 126, 1, 224, 108, 132, 120

Offset: 1

Antti Karttunen, Nov 06 2021

See comments in A348988.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A332994, A333784, A348987, A348988 (numerators).

Cf. also A348978, A348979.

f[n_] := n/FactorInteger[n][[-1, 1]]; g[1] = 1; g[n_] := g[n] = n + g[f[n]]; a[n_] := Denominator[g[n]/DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));
A348989(n) = { my(s=sigma(n)); (s/gcd(s, A332994(n))); };

a(n) = A000203(n) / A348987(n) = A000203(n) / gcd(A000203(n), A332994(n)).

A348977 a(n) = gcd(sigma(n), A332993(n)).

Original entry on oeis.org

1, 3, 4, 7, 6, 2, 8, 15, 13, 2, 12, 2, 14, 2, 3, 31, 18, 1, 20, 6, 1, 2, 24, 2, 31, 2, 40, 2, 30, 3, 32, 63, 3, 2, 1, 1, 38, 2, 1, 2, 42, 1, 44, 6, 6, 2, 48, 2, 57, 3, 3, 2, 54, 2, 1, 2, 1, 2, 60, 3, 62, 2, 4, 127, 1, 3, 68, 6, 3, 1, 72, 1, 74, 2, 2, 2, 1, 1, 80, 6, 121, 2, 84, 1, 1, 2, 3, 2, 90, 78, 7, 6, 1, 2, 5

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)

Cf. A000203, A332993, A333783, A348978, A348979.

Cf. also A348987.

f[n_] := n/FactorInteger[n][[1, 1]]; g[1] = 1; g[n_] := g[n] = n + g[f[n]]; a[n_] := GCD[g[n], DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A332993(n) = if(1==n,n,n + A332993(n/vecmin(factor(n)[,1])));
A348977(n) = gcd(sigma(n), A332993(n));

a(n) = gcd(A000203(n), A332993(n)).
a(n) = gcd(A000203(n), A333783(n)) = gcd(A332993(n), A333783(n)).
a(n) = A332993(n) / A348978(n) = A000203(n) / A348979(n).

cp's OEIS Frontend

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

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A348989 Denominator of A332994(n) / sigma(n).

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A348977 a(n) = gcd(sigma(n), A332993(n)).

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