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

1,3

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

Links

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

Formula

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

Example

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A096107 A329585 A128487 * A014673 A346704 A280686

Adjacent sequences: A056606 A056607 A056608 * A056610 A056611 A056612

Keyword

nonn

Author

Labos Elemer, Aug 07 2000

Extensions

Definition and example changed by Luc Rousseau, Jan 02 2019

Status

approved