cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

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

Crossrefs

Cf. A000203, A000961, A332994, A333784, A348987, A348989 (denominators).
Cf. also A348978.

Programs

  • 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 *)
  • 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)); };

Formula

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

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

Comments

See comments in A348988.

Links

Crossrefs

Cf. A000203, A332994, A333784, A348987, A348988 (numerators).
Cf. also A348978, A348979.

Programs

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

Formula

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

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

Links

Crossrefs

Cf. A000203, A332993, A333783, A348978, A348979.
Cf. also A348987.

Programs

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

Formula

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).

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

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

Comments

See comments in A348978.

Links

Crossrefs

Cf. A000203, A332993, A333783, A348977, A348978 (numerators).
Cf. also A348988, A348989.

Programs

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

Formula

a(n) = A000203(n) / A348977(n) = A000203(n) / gcd(A000203(n), A332993(n)).
Showing 1-4 of 4 results.

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