OFFSET
1,2
COMMENTS
For the corresponding sequence for numbers of the form k^3+1 instead of k^2+1, the only terms known to me are 0 and 26, with 26^3+1 = (2^3+1)^2*(6^3+1).
LINKS
David Trimas, Table of n, a(n) for n = 1..2260
EXAMPLE
0 is a term because 0^2+1 = 1 equals the empty product.
3 is a term because 3^2+1 = 10 = 2*5 = (1^2+1)*(2^2+1).
38 is a term because 38^2+1 = 1445 = 5*17*17 = (2^2+1)*(4^2+1)^2. (This is the first term that requires more than two factors.)
MATHEMATICA
g[lst_, p_] :=
Module[{t, i, j},
Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
multPartition[n_] :=
Module[{i, j, p, e, lst = {{}}}, {p, e} =
Transpose[FactorInteger[n]];
Do[lst = g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
output = Join[{0}, Flatten[Position[Table[
test = Sqrt[multPartition[n^2 + 1][[2 ;; All]] - 1];
Count[AllTrue[#, IntegerQ] & /@ test, True] > 0
, {n, 707}], True]]]
(* David Trimas, Jul 23 2023 *)
CROSSREFS
Sequences that list those terms (or their indices or some other key) of a given sequence that are products of smaller terms of the same sequence (in other words, the nonprimitive terms of the multiplicative closure of the sequence):
this sequence (A002522),
KEYWORD
nonn
AUTHOR
Pontus von Brömssen, Jun 19 2023
STATUS
approved