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

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

Author

Jonathan Sondow, Apr 28 2004

Keywords

• nonn

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