A072066 Exceptional (or extraordinary) numbers: m such that A005179(m) < A037019(m).

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

Offset: 1

Author

David Wasserman, Jun 12 2002

Keywords

• nonn

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

Only very few of the initial terms, {108, 162, 243, 324, 486, 729, ...} are not multiples of 8. Note that the 2nd to 6th in this list (and certainly more) equal 81*k = (10 + 1/8)*a(n) with n = 2, 3, 4, 5, 7, ... - M. F. Hasler, Jun 15 2022

Examples

			m=8 is a term: A005179(8) = 2^3 * 3 = 24 < 30 = 2^1 * 3^1 * 5^1 = A037019(8). - _Jon E. Schoenfield_, Mar 18 2022
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Ron Brown, The minimal number with a given number of divisors, Journal of Number Theory 116:1 (2006), pp. 150-158.
• M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
• Shu-Yuan Mei, A new class of ordinary integers, video summary of article.
• Shu-Yuan Mei, A new class of ordinary integers, Journal of Number Theory, Volume 133, Issue 10, October 2013, Pages 3559-3564.
• Anna K. Savvopoulou and Christopher M. Wedrychowicz, On the smallest number with a given number of divisors, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.

Crossrefs

Cf. A005179, A037019.

Programs

PARI

select( {is_A072066(n)=A005179(n)<A037019(n)}, [1..9999]) \\ M. F. Hasler, Oct 14 2014, updated Jun 15 2022

Extensions

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