A364151 -id:A364151 - 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 *)

A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

Original entry on oeis.org

1, 36, 1, 45, 560, 1, 210, 19600, 4900, 1, 300, 43680, 513590, 56448, 1, 378, 45760, 333833500, 127008, 4750, 1, 630, 893200, 711410700, 259200, 1926049000, 58372180608, 1

Offset: 2

Pontus von Brömssen, Jul 09 2024

If there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

The first term in each row is 1, because 1 is an n-gonal pyramidal number for every n and it equals the empty product.

			Array begins:
  n\k| 1     2          3            4
  ---+--------------------------------
  2  | 1    36         45          210
  3  | 1   560      19600        43680
  4  | 1  4900     513590    333833500
  5  | 1 56448     127008       259200
  6  | 1  4750 1926049000 655578709500

Cf. A080851, A374370, A374499 (second column).

Rows: A068143 (n=2 except the first term), A364151 (n=3), A374500 (n=4), A374501 (n=5), A374502 (n=6).

A374375 Positive numbers of the form k(k+1)(k+2) that are products of smaller numbers of that same form.

Original entry on oeis.org

720, 262080, 43243200, 85765680, 14366626560, 27680637600, 8916100427520, 2871098559070560, 5720836667515200, 20123426048544000, 924491486191094640, 297683700627082714560

Offset: 1

Pontus von Brömssen, Jul 07 2024

All terms are divisible by 36, because the number k*(k+1)*(k+2) is always divisible by 6 so a product of at least 2 such factors is divisible by 36. The first 12 terms are even divisible by 720.
a(13) > 4.5*10^22 if it exists. - David A. Corneth, Jul 12 2024
b(F(2*k)^2-1) is a term for all k >= 2, where b(k) = k*(k+1)*(k+2) = A007531(k+2) and F = A000045 is the Fibonacci sequence, because b(F(2*k)^2-1) = b(F(2*k-1)-1)*b(F(2*k+1)-1). In particular, a(13) <= b(F(20)^2-1) = 95853241822852400000400. - Pontus von Brömssen, Jul 13 2024

			With b(k) = k*(k+1)*(k+2) = A007531(k+2), we have the following factorizations of the first 12 terms:
                    720 =       b(8) = 6*120 = b(1)*b(4);
                 262080 =      b(63) = 120*2184 = b(4)*b(12);
               43243200 =     b(350) = 120*210*1716 = b(4)*b(5)*b(11);
               85765680 =     b(440) = 2184*39270 = b(12)*b(33);
            14366626560 =    b(2430) = 24*60*1716*5814 = b(2)*b(3)*b(11)*b(17);
            27680637600 =    b(3024) = 39270*704880 = b(33)*b(88);
          8916100427520 =   b(20735) = 704880*12649104 = b(88)*b(232);
       2871098559070560 =  b(142128) = 12649104*226980390 = b(232)*b(609);
       5720836667515200 =  b(178848) = 6*210*373176*12166770 = b(1)*b(5)*b(71)*b(229);
      20123426048544000 =  b(271998) = 6*210*328440*48626760 = b(1)*b(5)*b(68)*b(364);
     924491486191094640 =  b(974168) = 226980390*4073001576 = b(609)*b(1596);
  297683700627082714560 = b(6677055) = 4073001576*73087057560 = b(1596)*b(4180).

David A. Corneth, PARI program

Cf. A000045, A007531, A364151, A374374.

a(8)-a(11) from Michael S. Branicky, Jul 07 2024

a(12) from David A. Corneth, Jul 12 2024

A196568 Binomial coefficients of the form C(k, 3) that are products of two other binomial coefficients of the form C(k, 3).

Original entry on oeis.org

560, 19600, 43680, 893200, 1521520, 7207200, 29269240, 2845642800, 26595476600, 249466897680

Offset: 1

Kausthub Gudipati, Oct 04 2011

a(11), if it exists, > C(5*10^6, 3). - D. S. McNeil, Oct 26 2011

			     560 = C( 16, 3) = C( 8, 3) * C( 5, 3);
   19600 = C( 50, 3) = C(16, 3) * C( 7, 3);
   43680 = C( 65, 3) = C(14, 3) * C(10, 3);
  893200 = C(176, 3) = C(30, 3) * C(12, 3).

Erich Friedman, Problem of the month October 2011

Subsequence of A364151 (more than 2 factors allowed).

Cf. A188630 (analog for triangular numbers).

Name clarified by Pontus von Brömssen, Jun 22 2023

A364152 Least n-simplex number (i.e., number of the form C(m,n) = binomial(m,n), m >= n), that can be written as a product of two or more smaller n-simplex numbers, or 0 if no such number exists.

Original entry on oeis.org

4, 36, 560, 20475, 126

Offset: 1

Pontus von Brömssen, Jul 15 2023

When n = k^2+3*k+1 is in A028387, C(n+k+3,n) = C(n+1,n) * C(n+k+1,n), so 0 != a(n) <= C(n+k+3,n). It appears that equality holds (verified for 0 <= k <= 100). In particular, a(11) = C(16,11) = 4368, a(19) = C(25,19) = 177100, a(29) = C(36,29) = 8347680, a(41) = C(49,41) = 450978066, ... .
a(34) = 4923689695575 = C(50,34) = C(35,34)*C(47,34).
a(6) > 10^29 (unless a(6) = 0). - Pontus von Brömssen, Jul 14 2024

			a(1) =     4 = C( 4, 1) = C(2,1) * C(2,1).
a(2) =    36 = C( 9, 2) = C(4,2)^2.
a(3) =   560 = C(16, 3) = C(5,3) * C(8,3). (Also, C(16,3) = C(4,3)^2 * C(7,3)).
a(4) = 20475 = C(28, 4) = C(6,4) * C(15,4).
a(5) =   126 = C( 9, 5) = C(6,5) * C(7,5).

a(1)-a(3) are the first terms greater than 1 in A018252, A068143, and A364151, respectively.

Cf. A028387.

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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A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

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A374375 Positive numbers of the form k*(k+1)*(k+2) that are products of smaller numbers of that same form.

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A196568 Binomial coefficients of the form C(k, 3) that are products of two other binomial coefficients of the form C(k, 3).

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A364152 Least n-simplex number (i.e., number of the form C(m,n) = binomial(m,n), m >= n), that can be written as a product of two or more smaller n-simplex numbers, or 0 if no such number exists.

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A374375 Positive numbers of the form k(k+1)(k+2) that are products of smaller numbers of that same form.