A348977 -id:A348977 - cp's OEIS frontend

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

Antti Karttunen, Nov 06 2021

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).

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

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

Cf. also A348988, A348989.

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 *)

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

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

A348979 Denominator of ratio A332993(n) / sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 9, 1, 14, 1, 12, 8, 1, 1, 39, 1, 7, 32, 18, 1, 30, 1, 21, 1, 28, 1, 24, 1, 1, 16, 27, 48, 91, 1, 30, 56, 45, 1, 96, 1, 14, 13, 36, 1, 62, 1, 31, 24, 49, 1, 60, 72, 60, 80, 45, 1, 56, 1, 48, 26, 1, 84, 48, 1, 21, 32, 144, 1, 195, 1, 57, 62, 70, 96, 168, 1, 31, 1, 63, 1, 224, 108, 66, 40, 90

Offset: 1

Antti Karttunen, Nov 06 2021

See comments in A348978.

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

Cf. A000203, A332993, A333783, A348977, A348978 (numerators).

Cf. also A348988, A348989.

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 *)

A332993(n) = if(1==n,n,n + A332993(n/vecmin(factor(n)[,1])));
A348979(n) = { my(s=sigma(n)); (s/gcd(s, A332993(n))); };

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

A348987 a(n) = gcd(sigma(n), A332994(n)).

Original entry on oeis.org

1, 3, 4, 7, 6, 3, 8, 15, 13, 1, 12, 1, 14, 1, 1, 31, 18, 3, 20, 3, 1, 1, 24, 3, 31, 1, 40, 7, 30, 3, 32, 63, 1, 1, 1, 1, 38, 1, 1, 5, 42, 3, 44, 3, 2, 1, 48, 1, 57, 3, 1, 1, 54, 3, 1, 1, 1, 1, 60, 1, 62, 1, 4, 127, 1, 3, 68, 3, 1, 1, 72, 3, 74, 1, 2, 1, 1, 3, 80, 3, 121, 1, 84, 1, 1, 1, 1, 1, 90, 117, 1, 3, 1, 1, 1, 3

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

Index entries for sequences related to sigma(n)

Cf. A000203, A332994, A333784, A348988, A348989.

Cf. also A348977.

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 *)

A332994(n) = if(1==n,n,n + A332994(n/vecmax(factor(n)[,1])));
A348987(n) = gcd(sigma(n), A332994(n));

a(n) = gcd(A000203(n), A332994(n)).
a(n) = gcd(A000203(n), A333784(n)) = gcd(A332994(n), A333784(n)).
a(n) = A332994(n) / A348988(n) = A000203(n) / A348989(n).

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

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A348979 Denominator of ratio A332993(n) / sigma(n).

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A348987 a(n) = gcd(sigma(n), A332994(n)).

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