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%I #20 Mar 26 2021 06:31:49
%S 1,6,12,20,24,40,60,80,90,112,120,180,240,315,336,360,504,560,630,720,
%T 1008,1260,1680,2016,2240,2520,3360,4032,4480,5040,6720,8064,10080,
%U 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
%N Numbers with incrementally smallest ratio A002034(n)/n.
%C The factorials, where A002034(n!)/n! = 1/(n-1)!, appear to form a subsequence. The numbers A007672(a(n)) are small.
%C 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
%H A. J. Kempner, <a href="http://www.jstor.org/stable/2972639">Miscellanea</a>, Amer. Math. Monthly, 25 (1918), 201-210 [ See Section II, "Concerning the smallest integer m! divisible by a given integer n". ]
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/SmarandacheFunction.html">MathWorld: Smarandache Function</a>
%e 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.
%t (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)
%t 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 = 1; 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 *)
%Y Cf. A002034, A094372, A094404, A094634.
%K nonn
%O 1,2
%A _Jonathan Sondow_, Apr 28 2004
%E More terms from _Robert G. Wilson v_, May 15 2004
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