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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.
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%I #28 Aug 03 2024 12:02:17

%S 24,40,48,56,72,80,88,96,104,112,120,136,144,152,160,162,168,176,184,

%T 192,200,208,224,232,240,248,264,272,280,288,296,304,312,320,328,336,

%U 344,352,360,368,376,384,392,400,405,408,416,424,440,448,456,464,472

%N 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.

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

%C 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

%H Michel Marcus, <a href="/A048104/b048104.txt">Table of n, a(n) for n = 1..10000</a>

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

%t Select[Range[500], AnyTrue[(f = FactorInteger[#]), First[#1] > Last[#1] &] && AnyTrue[f, First[#1] < Last[#1] &] &] (* _Amiram Eldar_, Nov 13 2020 *)

%o (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

%Y Cf. A048102, A048103.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Reiner Martin_, Jul 07 2001