login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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.
3
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
1,1
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
LINKS
EXAMPLE
48 = 2^4*3^1 is a term but 12 = 2^2*3^1 is not.
MATHEMATICA
Select[Range[500], AnyTrue[(f = FactorInteger[#]), First[#1] > Last[#1] &] && AnyTrue[f, First[#1] < Last[#1] &] &] (* Amiram Eldar, Nov 13 2020 *)
PROG
(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
CROSSREFS
Sequence in context: A362148 A062374 A272593 * A334801 A362594 A360793
KEYWORD
nonn,easy
EXTENSIONS
More terms from Reiner Martin, Jul 07 2001
STATUS
approved