A309544 - OEIS

%I #17 Feb 10 2023 04:52:44

%S 1,3,5,6,8,11,12,14,20,24,25,38,54,62,80,90,110,138,150,164,168,192,

%T 194,272,278,314,332,348,398,402,434,500,572,642,644,720,728,733,762,

%U 798,812,860,864,878,920,992,1013,1020,1022,1070,1092,1098,1118,1130,1182,1202,1230,1260,1308,1413,1434

%N Numbers k such that A001414(k^3-1) is divisible by k.

%C Contains k such that k-1 and k^2+k+1 are primes. Numbers in the sequence that are not of this form include 1, 5, 11, 25, 733, 1013, 1413, 6289, 16456, and 161307. Are there infinitely many of these?

%H Robert Israel, <a href="/A309544/b309544.txt">Table of n, a(n) for n = 1..10000</a>

%e 5 is a term because the prime factorization of 5^3-1 = 124 is 2^2*31 and 2+2+31=35 is divisible by 5.

%p filter:= proc(n) local F, t, y;

%p   F:= ifactors(n^3-1)[2];

%p   y:= add(t[1]*t[2], t=F);

%p   y mod n = 0

%p end proc:

%p select(filter, [$1..2000]);

%t sopfr[n_] := Total[Times @@@ FactorInteger[n]];

%t okQ[k_] := Divisible[sopfr[k^3-1], k];

%t Select[Range[2000], okQ] (* _Jean-François Alcover_, Feb 10 2023 *)

%Y Cf. A001414, A309534, A309542.

%K nonn

%O 1,2

%A _J. M. Bergot_ and _Robert Israel_, Aug 06 2019