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A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

%I A363636 #35 Jul 11 2024 10:47:07
%S A363636 0,3,7,13,17,18,21,31,38,43,47,57,68,73,91,99,111,117,123,132,133,157,
%T A363636 183,211,241,242,253,255,268,273,293,302,307,313,322,327,343,381,413,
%U A363636 421,438,443,463,487,507,515,553,557,577,593,601,651,693,697,703,707
%N A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.
%C A363636 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).
%H A363636 David Trimas, <a href="/A363636/b363636.txt">Table of n, a(n) for n = 1..2260</a>
%e A363636 0 is a term because 0^2+1 = 1 equals the empty product.
%e A363636 3 is a term because 3^2+1 = 10 = 2*5 = (1^2+1)*(2^2+1).
%e A363636 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.)
%t A363636 g[lst_, p_] :=
%t A363636   Module[{t, i, j},
%t A363636    Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
%t A363636       Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
%t A363636     Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
%t A363636 multPartition[n_] :=
%t A363636   Module[{i, j, p, e, lst = {{}}}, {p, e} =
%t A363636     Transpose[FactorInteger[n]];
%t A363636    Do[lst = g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
%t A363636 output = Join[{0}, Flatten[Position[Table[
%t A363636      test = Sqrt[multPartition[n^2 + 1][[2 ;; All]] - 1];
%t A363636      Count[AllTrue[#, IntegerQ] & /@ test, True] > 0
%t A363636      , {n, 707}], True]]]
%t A363636 (* _David Trimas_, Jul 23 2023 *)
%Y A363636 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):
%Y A363636   A018252 (A000027),
%Y A363636   A034878 (A000142),
%Y A363636   A068143 (A000217),
%Y A363636   A363492 (A000041),
%Y A363636   A363634 (A000959),
%Y A363636   A363635 (A003309),
%Y A363636   this sequence (A002522),
%Y A363636   A363637 (A005563),
%Y A363636   A363638 (A008864),
%Y A363636   A363750 (A006093),
%Y A363636   A364151 (A000292),
%Y A363636   A374372 (A000326),
%Y A363636   A374373 (A000384),
%Y A363636   A374374 (A002378),
%Y A363636   A374375 (A007531),
%Y A363636   A374500 (A000330),
%Y A363636   A374501 (A002411),
%Y A363636   A374502 (A002412).
%K A363636 nonn
%O A363636 1,2
%A A363636 _Pontus von Brömssen_, Jun 19 2023