A363634 -id:A363634 - cp's OEIS frontend

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.

0, 3, 7, 13, 17, 18, 21, 31, 38, 43, 47, 57, 68, 73, 91, 99, 111, 117, 123, 132, 133, 157, 183, 211, 241, 242, 253, 255, 268, 273, 293, 302, 307, 313, 322, 327, 343, 381, 413, 421, 438, 443, 463, 487, 507, 515, 553, 557, 577, 593, 601, 651, 693, 697, 703, 707

Offset: 1

Pontus von Brömssen, Jun 19 2023

nonn

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).

			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.)

David Trimas, Table of n, a(n) for n = 1..2260

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):
  A018252 (A000027),
  A034878 (A000142),
  A068143 (A000217),
  A363492 (A000041),
  A363634 (A000959),
  A363635 (A003309),
  this sequence (A002522),
  A363637 (A005563),
  A363638 (A008864),
  A363750 (A006093),
  A364151 (A000292),
  A374372 (A000326),
  A374373 (A000384),
  A374374 (A002378),
  A374375 (A007531),
  A374500 (A000330),
  A374501 (A002411),
  A374502 (A002412).

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 *)

A363635 Ludic numbers that are products of smaller ludic numbers.

Original entry on oeis.org

1, 25, 77, 91, 115, 119, 121, 143, 161, 175, 221, 235, 265, 287, 301, 329, 377, 407, 415, 445, 481, 493, 497, 517, 535, 581, 595, 625, 667, 697, 749, 805, 841, 851, 865, 913, 943, 1015, 1043, 1045, 1105, 1177, 1207, 1225, 1247, 1351, 1363, 1375, 1391, 1403

Offset: 1

Pontus von Brömssen, Jun 19 2023

nonn

			1 is a term because it is a ludic number and equals the empty product.
25 is a term because 25 = 5*5 and both 25 and 5 are ludic numbers.
1015 is a term because 1015 = 5*7*29 and both 1015 and the three factors 5, 7, and 29 are ludic numbers. (This is the first term that requires more than two factors.)

Cf. A003309, A363634, A363636.

cp's OEIS Frontend

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.

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