A048104 If n = Product p_i^e_i (e_i >= 1) then for some i, p_i > e_i and for some j, p_j < e_j.

24, 40, 48, 56, 72, 80, 88, 96, 104, 112, 120, 136, 144, 152, 160, 162, 168, 176, 184, 192, 200, 208, 224, 232, 240, 248, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 405, 408, 416, 424, 440, 448, 456, 464, 472

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The asymptotic density of this sequence is 1 - Product_{p prime} (1-1/p^(p+1)) = 0.13585792767780221591... . - Amiram Eldar, Feb 14 2023

Verified up to a(120) = 1000, except for a(16) = 162 and a(55) = 486, every a(n) is also the order of an isomorphism class for which there exists at least one nonabelian nilpotent group G such that |Aut(G)|/a(n) is nonintegral. Within the same range there are 26 group orders not in a(n), which, except for 3^4*2^3 = 648, all have the form 3^3*m or 5^3*k, with m and k being prime, squarefree, or nonsquarefree. - Miles Englezou, Jul 16 2024

Examples

			48 = 2^4*3^1 is a term but 12 = 2^2*3^1 is not.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A048102, A048103.

Programs

Mathematica

Select[Range[500], AnyTrue[(f = FactorInteger[#]), First[#1] > Last[#1] &] && AnyTrue[f, First[#1] < Last[#1] &] &] (* Amiram Eldar, Nov 13 2020 *)

PARI

isok(n) = my(f=factor(n), b1=0, b2=0); for (i=1, #f~, if (f[i,1] < f[i,2], b1=1, if (f[i,1] > f[i,2], b2=1))); return(b1 && b2); \\ Michel Marcus, Nov 13 2020

Extensions

More terms from Reiner Martin, Jul 07 2001