A093455 -id:A093455 - cp's OEIS frontend

A093456 Product of composite numbers among next n numbers.

1, 1, 24, 720, 2520, 120960, 259459200, 1357171200, 4929724800, 42608389824000, 11912739135897600, 59907396092544000, 20458385028297216000, 7926428532945162240000, 4693751193479184764928000, 328774885640356760904499200000, 12797917159224592605450240000

Offset: 1

Amarnath Murthy, Apr 03 2004

Conjecture: There are finitely many numbers such that a(n) is not == 0 (mod a(n-1)). (Also mentioned in A093455.)

Product of all composite numbers between n*(n-1)/2+1 and n*(n+1)/2 (including boundaries). - Stefan Steinerberger, Apr 02 2006

			Sequence begins:
   1:             a(1) = 1.
   2  3:          a(2) = 1.
   4  5  6:       a(3) = 4*6 = 24.
   7  8  9 10:    a(4) = 8*9*10 = 720.
  11 12 13 14 15: a(5) = 12*14*15 = 2520.
  ...

Harvey P. Dale, Table of n, a(n) for n = 1..244

Cf. A000217, A057003, A093455, A093457.

Table[a := Range[n*(n - 1)/2 + 1, n*(n + 1)/2]; b := Select[a, Not[PrimeQ[ # ]] &]; Product[b[[i]], {i, 1, Length[b]}], {n, 1, 20}] (* Stefan Steinerberger, Apr 02 2006 *)
Module[{nn=20},Times@@Select[#,CompositeQ]&/@TakeList[Range[(nn(nn+1))/2],Range[nn]]] (* Harvey P. Dale, Dec 30 2024 *)

a(n) = A057003(n)/A093457(n). - Michel Marcus, Jan 14 2025

More terms from Stefan Steinerberger, Apr 02 2006

A093457 Product of primes in the range [T(n-1) + 1, T(n - 1) + n], where T(n) is the n-th triangular number.

Original entry on oeis.org

1, 6, 5, 7, 143, 323, 23, 899, 65231, 2491, 3599, 347261, 583573, 1009091, 1317919, 16637, 428448457, 4273697, 5605027, 1445140189, 2445956099, 3368562317, 4927316309, 6454166203, 9473323417, 13508676341, 17347785757, 8989229423381

Offset: 1

Amarnath Murthy, Apr 03 2004

nonn

			a(5) = 11*13 = 143.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A093455, A093456.

Do[k = 1; Do[If[PrimeQ[i], k = k*i], {i, Binomial[n, 2] + 1, Binomial[n, 2] + n}]; Print[k], {n, 1, 30}] (* Ryan Propper, Jun 22 2005 *)
Join[{1,6},Table[With[{t=(n(n+1))/2},Times@@Select[Range[t+1,t+n],PrimeQ]],{n,2,30}]]  (* Harvey P. Dale, Jan 18 2025 *)

More terms from Ryan Propper, Jun 22 2005

cp's OEIS Frontend

A093456 Product of composite numbers among next n numbers.

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