A331613 - OEIS

%I #14 Jan 25 2025 03:15:46

%S 243,729,1215,2187,3645,6561,10935,15309,19683,32805,45927,54675,

%T 59049,98415,137781,164025,177147,216513,255879,273375,295245,334611,

%U 373977,413343,452709,492075,531441,570807,610173,649539,688905,728271,767637,807003,820125,846369,885735,925101

%N Odd exceptional numbers: odd k such that A005179(k) < A037019(k).

%C This sequence is infinite, because 3^p is a term for all p >= 5.

%C It seems that the smallest p-rough exceptional number (i.e., the smallest exceptional number whose smallest prime factor is p) is p^k, where k is the smallest number such that prime(k) > 2^p (p = 2 gives 2^3 = 8, p = 3 gives 3^5 = 243, p = 5 gives 5^12 = 244140625, ...).

%H Amiram Eldar, <a href="/A331613/b331613.txt">Table of n, a(n) for n = 1..256</a>

%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.

%e The smallest number with 243 divisors is 2^8 * 3^2 * 5^2 * 7^2 = 2822400, while A037019(243) = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 = 5336100 > A005179(243), so 243 is a term.

%o (PARI) isA331613(n) = (n%2) && A037019(n) > A005179(n) \\ See A005179 and A037019 for their programs

%Y Cf. A072066 (exceptional numbers), A005179, A037019.

%K nonn

%O 1,1

%A _Jianing Song_, Jan 22 2020