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%I #19 Feb 14 2021 12:58:44
%S 2,5,7,10,13,16,19,20,31,34,37,40,41,44,46,48,51,53,55,56,62,64,65,67,
%T 68,69,73,74,76,79,81,82,83,84,85,86,87,89,95,96,97,99,100,101,106,
%U 107,108,109,111,112,115,119,121,122,123,124,133,134,135,136,137,138,143
%N Numbers k such that Sum_{j=1..k} A008472(j) divides k!.
%H Giovanni Resta, <a href="/A067008/b067008.txt">Table of n, a(n) for n = 1..10000</a> (first 4114 terms from Daniel Starodubtsev)
%F a(n) seems to be asymptotic to c*n with c=2.3.....
%t tot[n_] := tot[n] = If[n < 2, 0, tot[n - 1] + Plus @@ FactorInteger[n][[;; , 1]]]; ef[n_, p_] := Block[{c=0, m=n}, While[m > 0, m = Floor[m/p]; c += m]; c]; ok[n_] := n > 1 && AllTrue[ FactorInteger@ tot@n, #[[1]] <= n && ef[n, #[[1]]] >= #[[2]] &]; Select[ Range@ 1000, ok] (* _Giovanni Resta_, Dec 13 2019 *)
%o (PARI) for (j=2,143,if(!(j!%sum(k=1,j,vecsum(factor(k)[,1]))),print1(j,", "))) \\ _Hugo Pfoertner_, Dec 13 2019
%Y Cf. A008472.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Oct 03 2002
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