A056609 a(n) = rad(n!)/rad(A001142(n)) where rad(n) is the squarefree kernel of n, A007947(n).

1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 1, 1, 7, 5, 2, 1, 3, 1, 5, 7, 11, 1, 1, 5, 13, 3, 7, 1, 1, 1, 2, 11, 17, 7, 1, 1, 19, 13, 1, 1, 7, 1, 11, 1, 23, 1, 1, 7, 5, 17, 13, 1, 3, 11, 1, 19, 29, 1, 1, 1, 31, 1, 2, 13, 11, 1, 17, 23, 1, 1, 1, 1, 37, 5, 19, 11, 13, 1, 1, 3, 41, 1, 1, 17, 43, 29, 11, 1, 1, 13

Offset: 1

Author

Labos Elemer, Aug 07 2000

Keywords

• nonn

Comments

The previous name, which does not match the data as observed by Luc Rousseau, was: Quotient of squarefree kernels of A002944(n) and A001405.

a(n) is the unique prime p not greater than n missing in the prime factorization of A001142(n), if such a prime exists; a(n) is 1 otherwise. - Luc Rousseau, Jan 01 2019

Examples

			From _Luc Rousseau_, Jan 02 2019: (Start)
In Pascal's triangle,
- row n=3 (1 3 3 1) contains no number with prime factor 2, so a(3) = 2;
- row n=4 (1 4 6 4 1) contains, for all p prime <= 4, a multiple of p, so a(4) = 1;
- row n=5 (1 5 10 10 5 1) contains no number with prime factor 3, so a(5) = 3;
etc.
(End)
		

Links

• Luc Rousseau, Table of n, a(n) for n = 1..1000 (first 90 terms from Labos Elemer)

Crossrefs

Cf. A002944, A001405, A003418, A002110, A001142, A034386, A056606.

Programs

Mathematica

L[n_] := Table[Binomial[n, k], {k, 1, Floor[n/2]}]
c[n_] := Complement[Prime /@ Range[PrimePi[n]], First /@ FactorInteger[Times @@ L[n]]]
a[n_] := Module[{x = c[n]}, If[x == {}, 1, First[x]]]
Table[a[n], {n, 1, 100}]
(* Luc Rousseau, Jan 01 2019 *)

PARI

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
b(n) = prod(m=1, n, binomial(n, m)); \\ A001142
a(n) = rad(n!)/rad(b(n)); \\ Michel Marcus, Jan 02 2019

Formula

a(n) = A034386(n) / A056606(n). - Sean A. Irvine, Apr 24 2022

Extensions

Definition and example changed by Luc Rousseau, Jan 02 2019