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 1-3 of 3 results.

A094371 Numbers with incrementally smallest ratio A002034(n)/n.

Original entry on oeis.org

1, 6, 12, 20, 24, 40, 60, 80, 90, 112, 120, 180, 240, 315, 336, 360, 504, 560, 630, 720, 1008, 1260, 1680, 2016, 2240, 2520, 3360, 4032, 4480, 5040, 6720, 8064, 10080, 12096, 12960, 13440, 17280, 18144, 20160, 24192, 25920, 30240, 34560, 36288, 40320, 51840, 60480, 72576, 86400, 90720, 103680, 113400, 120960, 145152, 151200, 172800, 181440, 226800, 241920, 259200, 295680, 302400
Offset: 1

Views

Author

Jonathan Sondow, Apr 28 2004

Keywords

Comments

The factorials, where A002034(n!)/n! = 1/(n-1)!, appear to form a subsequence. The numbers A007672(a(n)) are small.
a(n) is either even, 19k, 23k or R*k, where R is a repunit prime. For example at 2.19.23=874, the corresponding repunit is divisible by 3 repunit primes. - Labos Elemer, Jun 04 2004

Examples

			The first 5 incrementally smallest ratios A002034(n)/n are 1, 1/2, 1/3, 1/4, 1/6. They occur at n = 1, 6, 12, 20, 24.

Links

  • A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210 [ See Section II, "Concerning the smallest integer m! divisible by a given integer n". ]
  • J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function

Crossrefs

Cf. A002034, A094372, A094404, A094634.

Programs

  • Mathematica
    (A002034[n_] := (m=1; While[ !IntegerQ[m!/n], m++ ]; m); M = {}; L = {}; Do[With[{s = A002034[n]}, If[s/n < Min[M], M = Append[M, s/n]; L = Append[L, n]]], {n, 100}]; L)
A002034[1] := 1; A002034[n_] := Max[A002034 @@@ FactorInteger[n]]; A002034[p_, 1] := p; A002034[p_, alpha_] := A002034[p, alpha] = Module[{a, k, r, i, nu, k0 = alpha(p - 1)}, i = nu = Floor[Log[p, 1 + k0]]; a[1] = 1; a[n_] := (p^n - 1)/(p - 1); k[nu] = Quotient[alpha, a[nu]]; r[nu] = alpha - k[nu]a[nu]; While[r[i] > 0, k[i - 1] = Quotient[r[i], a[i - 1]]; r[i - 1] = r[i] - k[i - 1]a[i - 1]; i-- ]; k0 + Plus @@ k /@ Range[i, nu]]; L = M = {}; a = Infinity; Do[ s = A002034[n]; If[s/n < a, a = s/n; AppendTo[M, a]; AppendTo[L, n]], {n, 40320}]; L (* Eric W. Weisstein, May 17 2004 *)

Extensions

More terms from Robert G. Wilson v, May 15 2004

A094372 Denominators of incrementally smallest ratios A002034(n)/n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 40, 15, 16, 24, 30, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 252, 280, 360, 420, 504, 560, 720, 840, 1008, 1260, 1344, 1440, 1680, 1920, 2016, 2520, 2688, 2880, 3360, 3456, 4032, 5040, 5760, 6720, 8064, 8640, 10080, 10368, 11340, 13440, 72576, 15120, 17280, 20160, 22680, 24192, 25920, 26880, 30240
Offset: 1

Views

Author

Jonathan Sondow, Apr 28 2004

Keywords

Comments

Numerators are in A094404. Same as A094371(n)/gcd(A094371(n), A002034(A094371(n))). The factorials appear to form a subsequence.

Examples

			a(5) = 6 because the 5th incrementally smallest ratio A002034(n)/n is 4/24 = 1/6.

Links

Crossrefs

Cf. A002034, A094371, A094404, A094634.

Programs

  • Mathematica
    (A002034[n_] := (m = 1; While[ !IntegerQ[m!/n], m++ ]; m); M = {}; Do[With[{s = A002034[n]}, If[s/n < Min[M], M = Append[M, s/n]]], {n, 120}]; Denominator[M])
A002034[1] := 1; A002034[n_] := Max[A002034 @@@ FactorInteger[n]]; A002034[p_, 1] := p; A002034[p_, alpha_] := A002034[p, alpha] = Module[{a, k, r, i, nu, k0 = alpha(p - 1)}, i = nu = Floor[Log[p, 1 + k0]]; a[1] = 1; a[n_] := (p^n - 1)/(p - 1); k[nu] = Quotient[alpha, a[nu]]; r[nu] = alpha - k[nu]a[nu]; While[r[i] > 0, k[i - 1] = Quotient[r[i], a[i - 1]]; r[i - 1] = r[i] - k[i - 1]a[i - 1]; i-- ]; k0 + Plus @@ k /@ Range[i, nu]]; M = {}; a = Infinity; Do[ s = A002034[n]; If[s/n < a, a = s/n; AppendTo[M, a]], {n, 2, 40320}]; Denominator[M] (* Eric W. Weisstein, May 17 2004 *)

Extensions

More terms from Robert G. Wilson v, May 15 2004

A094404 Numerators of low-water marks of mu(n)/n, where mu(n) is A002034.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Eric W. Weisstein, Apr 29 2004

Keywords

Comments

Denominators are A094372 and positions are A094371.

Examples

			1, 1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 3/40, 1/15, 1/16, ...

Links

Crossrefs

Cf. A002034, A094371, A094372, A094634.

Programs

  • Mathematica
    A002034[1] := 1; A002034[n_] := Max[A002034 @@@ FactorInteger[n]]; A002034[p_, 1] := p; A002034[p_, alpha_] := A002034[p, alpha] = Module[{a, k, r, i, nu, k0 = alpha(p - 1)}, i = nu = Floor[Log[p, 1 + k0]]; a[1] = 1; a[n_] := (p^n - 1)/(p - 1); k[nu] = Quotient[alpha, a[nu]]; r[nu] = alpha - k[nu]a[nu]; While[r[i] > 0, k[i - 1] = Quotient[r[i], a[i - 1]]; r[i - 1] = r[i] - k[i - 1]a[i - 1]; i-- ]; k0 + Plus @@ k /@ Range[i, nu]]; M = {}; a = Infinity; Do[ s = A002034[n]; If[s/n < a, a = s/n; AppendTo[M, a]], {n, 40320}]; Numerator[M] (* Jonathan Sondow, Apr 28 2004, revised by Eric W. Weisstein, May 17 2004 *)
Showing 1-3 of 3 results.

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