A072066 - OEIS

%I #43 Jun 17 2022 03:22:31

%S 8,16,24,32,48,64,72,80,96,108,112,128,144,160,162,176,192,208,216,

%T 224,243,256,272,288,304,320,324,352,368,384,416,432,448,464,480,486,

%U 496,512,544,576,592,608,640,648,656,672,688,704,729,736,752,768,832,848

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

%C 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

%C Not a subsequence of A025487: 80, 108, 112, etc. are not the product of primorials. - _Charles R Greathouse IV_, Jun 07 2013

%C The product of any exceptional numbers is an exceptional number. - _Thomas Ordowski_, Jun 14 2015

%C 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

%C 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

%H T. D. Noe, <a href="/A072066/b072066.txt">Table of n, a(n) for n = 1..1000</a>

%H Ron Brown, <a href="http://dx.doi.org/10.1016/j.jnt.2005.04.004">The minimal number with a given number of divisors</a> (2009), Journal of Number Theory 116:1 (2005), pp. 150-158.

%H M. E. Grost, <a href="http://www.jstor.org/stable/2315183">The smallest number with a given number of divisors</a>, Amer. Math. Monthly, 75 (1968), 725-729.

%H Shu-Yuan Mei, <a href="https://www.youtube.com/watch?v=WTY4wr8L_U0">A new class of ordinary integers</a>, video summary of article.

%H Shu-Yuan Mei, <a href="http://dx.doi.org/10.1016/j.jnt.2013.04.019">A new class of ordinary integers</a>, Journal of Number Theory, Volume 133, Issue 10, October 2013, Pages 3559-3564.

%H Anna K. Savvopoulou and Christopher M. Wedrychowicz, <a href="http://dx.doi.org/10.1007/s11139-014-9572-9">On the smallest number with a given number of divisors</a>, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.

%e 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

%o (PARI) select( {is_A072066(n)=A005179(n)<A037019(n)}, [1..9999]) \\ _M. F. Hasler_, Oct 14 2014, updated Jun 15 2022

%Y Cf. A005179, A037019.

%K nonn

%O 1,1

%A _David Wasserman_, Jun 12 2002

%E Links updated by _Michel Marcus_ and _M. F. Hasler_, Oct 14 2014