Weird Sequence of Positive Integers (FWD)

[David W. Wilson]

Antreas P. Hatzipolakis wrote:

> >From: w_harden at bellsouth.net (David Harden)
> >Newsgroups: sci.math
> >Subject: Weird Sequence of Positive Integers
> >Date: Sun, 06 Jun 1999 19:16:08 GMT
> >

> [ description of sequence elided ]

> >In general, for what positive integers k does there always exist a
> >positive integer n such that for any prime p p^k divides n! but
> >p^(k+1) does not? The first few values of k like this are 1, 4, 8, 10,
> >18, 22, 26, 32, 34, 46, 49, 50, 57, 66, 70, 74, 81, 82, 94, 102, 130,
> >134, 138, 142, 152, 162, 165, 166, 174, 176, 183, 184, 201, 205, 206,
> >222, 231, 232, 236, 237, 244, 246, and 256.

A good description of this sequence is:

"Every prime occurs to this power in some factorial".Up to 1000, the sequence goes

0 1 4 8 10 18 22 26 32 34 46 49 50 57 66 70 74 81 82 86 94 102 130 134
138 142 152 162 165 166 174 176 183 184 201 205 206 222 231 232 236 237
244 246 256 270 273 274 286 290 296 304 312 318 326 328 343 344 352 356
368 374 382 386 390 392 405 408 436 438 445 448 452 471 476 502 526 534
542 546 565 574 582 592 596 606 608 616 628 638 642 656 669 687 694 704
705 717 742 750 754 758 770 774 776 782 785 800 808 812 822 830 836 844
868 875 876 894 908 924 925 931 939 943 946 950 957 963 964 972 988

0 belongs in the sequence since every prime occurs to power 0 in 1! = 1.
Though this sequence ostensibly grows like the primes, it contains no
primes, as no prime p occurs to power p in any factorial.