A222759 Conjectured number of primes p for which binomial(n*p,p) (mod p^3) does not equal n.

0, 2, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

T. D. Noe, Mar 12 2013

Keywords

• nonn

Comments

It appears that, for k > 2 and n >= prime(prime(k)^3), then a(n) >= k.

Sequences A000720 and A056811 give results for binomial(n*p,p) (mod p) and binomial(n*p,p) (mod p^2), respectively. It appears that mod p^3 is the last case; that is, this identity does not hold for higher powers. - T. D. Noe, Mar 14 2013

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A096328 (prime(prime(n)^3)).

Cf. A000720, A056811 (primePi(n) and primePi(sqrt(n))).

Programs

Mathematica

Table[Length[Select[Prime[Range[100]], Mod[Binomial[n*#,#], #^3] != n &]], {n, 87}]