A309544 Numbers k such that A001414(k^3-1) is divisible by k.

1, 3, 5, 6, 8, 11, 12, 14, 20, 24, 25, 38, 54, 62, 80, 90, 110, 138, 150, 164, 168, 192, 194, 272, 278, 314, 332, 348, 398, 402, 434, 500, 572, 642, 644, 720, 728, 733, 762, 798, 812, 860, 864, 878, 920, 992, 1013, 1020, 1022, 1070, 1092, 1098, 1118, 1130, 1182, 1202, 1230, 1260, 1308, 1413, 1434

Offset

1,2

Comments

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?

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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.

Maple

filter:= proc(n) local F, t, y;
  F:= ifactors(n^3-1)[2];
  y:= add(t[1]*t[2], t=F);
  y mod n = 0
end proc:
select(filter, [$1..2000]);

Mathematica

sopfr[n_] := Total[Times @@@ FactorInteger[n]];
okQ[k_] := Divisible[sopfr[k^3-1], k];
Select[Range[2000], okQ] (* Jean-François Alcover, Feb 10 2023 *)

Crossrefs

Cf. A001414, A309534, A309542.

Sequence in context: A131422 A301744 A087720 * A154111 A047444 A279933

Adjacent sequences: A309541 A309542 A309543 * A309545 A309546 A309547

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Aug 06 2019

Status

approved