A133481 a(1) = 1; for n > 1, a(n) is the least k such that k^n divides k! but k^(n+1) does not divide k!.

1, 6, 15, 18, 12, 32, 24, 36, 40, 45, 48, 100, 84, 60, 154, 165, 72, 96, 80, 126, 90, 135, 286, 200, 312, 264, 168, 120, 297, 189, 160, 330, 544, 210, 144, 224, 300, 385, 396, 324, 252, 680, 350, 180, 280, 748, 572, 486, 400, 405, 315, 528, 320, 336, 450, 512, 288, 240, 715

Offset: 1

Author

Masahiko Shin, Nov 29 2007

Keywords

• nonn
• look
• nice

Comments

Least k such that A011776(k) = n.

New record highs, by index: 1, 2, 3, 4, 6, 8, 9, 10, 11, 12, 15, 16, 23, 25, 32, 33, 42, 46, 63, 66, 79, 85, 100, 119, 128, 167, 188, 201, 213, 226, 240, 256, 335, 346, 348, 352, 360, 377, 385, 414, 426, 480, 481, 494, 504, 533, 555, 596, 656, 727, 883, 926, 938, 1026, 1094, ... - Robert G. Wilson v, Feb 28 2012

First 10000 terms are 163-smooth. - David A. Corneth, Mar 15 2019

Examples

			a(7)=24 because 24^7|24! and smaller numbers than 24 do not divide their factorials 7 times.
a(2) = 6 as 6^2|6! but 6! doesn't divide 6^(2 + 1) and 6 is the least positive integer with this property. - _David A. Corneth_, Mar 15 2019
		

References

• Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.
• J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 251.

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (terms 1..345 from T. D. Noe, terms 346..1150 from Robert G. Wilson v)
• David A. Corneth, PARI program.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A011776, A011777, A011778.

Programs

Mathematica

kdn[n_]:=Module[{k=2},While[!Divisible[k!,k^n]||Divisible[k!, k^(n+1)], k++];k]; Join[{1},Array[kdn,60,2]] (* Harvey P. Dale, Feb 27 2012 *)

PARI

a(n)=if(n<2,1,my(k=2);while(valuation(k!,k)!=n,k++);k) \\ Charles R Greathouse IV, Feb 27 2012

PARI

See Corneth link \\ David A. Corneth, Mar 15 2019

Extensions

Edited by N. J. A. Sloane using material from A011777, Nov 29 2007