

A072066


Exceptional (or extraordinary) numbers: n such that A005179(n) < A037019(n).


10



8, 16, 24, 32, 48, 64, 72, 80, 96, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 480, 486, 496, 512, 544, 576, 592, 608, 640, 648, 656, 672, 688, 704, 729, 736, 752, 768, 832, 848
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OFFSET

1,1


COMMENTS

Brown shows that this sequence has density 0 and is a subsequence of A013929. Mei shows that in fact it is a subsequence of A048108.  Charles R Greathouse IV, Jun 07 2013
Not a subsequence of A025487: 80, 108, 112, etc. are not the product of primorials.  Charles R Greathouse IV, Jun 07 2013
The product of any exceptional numbers is an exceptional number.  Thomas Ordowski, Jun 14 2015
Grost proved that p^k is in the sequence if and only if 2^p < prime(k), where p is a prime.  Thomas Ordowski, Jun 15 2015


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
Ron Brown, The minimal number with a given number of divisors (2009), Journal of Number Theory 116:1 (2005), pp. 150158.
M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725729.
ShuYuan Mei, A new class of ordinary integers, video summary of article.
ShuYuan Mei, A new class of ordinary integers, Journal of Number Theory, Volume 133, Issue 10, October 2013, Pages 35593564.
Anna K. Savvopoulou and Christopher M. Wedrychowicz, On the smallest number with a given number of divisors, The Ramanujan Journal, 2015, Vol. 37, pp. 5164.


PROG

(PARI) for(n=1, 9999, A005179(n)<A037019(n)&&print1(n", ")) \\ M. F. Hasler, Oct 14 2014


CROSSREFS

Cf. A005179, A037019.
Sequence in context: A020335 A044833 A033005 * A342018 A055065 A181311
Adjacent sequences: A072063 A072064 A072065 * A072067 A072068 A072069


KEYWORD

nonn


AUTHOR

David Wasserman, Jun 12 2002


EXTENSIONS

Links updated by Michel Marcus and M. F. Hasler, Oct 14 2014


STATUS

approved



