A096328 -id:A096328 - cp's OEIS frontend

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

T. D. Noe, Mar 12 2013

nonn

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

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

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

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

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

A243896 a(n) = prime(n^2+1).

Original entry on oeis.org

2, 3, 11, 29, 59, 101, 157, 229, 313, 421, 547, 673, 829, 1013, 1201, 1429, 1621, 1889, 2153, 2441, 2749, 3089, 3463, 3821, 4217, 4639, 5059, 5521, 6011, 6491, 7001, 7577, 8167, 8741, 9343, 9941, 10631, 11329, 12071, 12757, 13513, 14341, 15107, 15881

Offset: 0

Freimut Marschner, Jun 17 2014

nonn

For n>1, the numbers prime(n^2-1), prime(n^2) and prime(n^2+1), that is, A243895(n), A001248(n) and a(n), constitute a triple of successive prime numbers.

			n = 4, n^2 = 16, n^2 + 1 = 17, prime(17) = 59.

Freimut Marschner, Table of n, a(n) for n = 0..97

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2). A011757 (prime(n^2)), A055875 (prime(n^3)), A096327 (prime((prime(n)^2))), A096328 (prime(prime(n)^3)), A038580 (prime(prime(prime(n)))).

Table[Prime[n^2+1],{n,0,50}] (* Harvey P. Dale, Dec 25 2022 *)

a(n) = prime(n^2 + 1) = prime(A000290(n) + 1) = prime(A002522(n)).

A243895 a(n) = prime(n^2-1).

Original entry on oeis.org

5, 19, 47, 89, 149, 223, 307, 409, 523, 659, 823, 997, 1187, 1423, 1613, 1877, 2141, 2423, 2731, 3079, 3457, 3797, 4201, 4621, 5039, 5507, 5987, 6473, 6991, 7561, 8147, 8731, 9337, 9929, 10613, 11317, 12043, 12739, 13487, 14323, 15091, 15859, 16741

Offset: 2

Freimut Marschner, Jun 17 2014

nonn

The prime numbers prime(k-1), prime(k) = A001248 and prime(k+1) = A243896 with k = n^2 are building a triple of successive prime numbers. Remark: prime(n^2-1) is not defined for n=1.

			n = 3, n^2 = 9, n^2-1 = 8, prime(8) = 19.

Freimut Marschner, Table of n, a(n) for n = 2..97

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A011757 (prime(n^2)), A055875 (prime(n^3)), A096327 (prime((prime(n)^2))), A096328 (prime(prime(n)^3)), A038580 (prime(prime(prime(n)))).

Table[Prime[n^2-1],{n,2,50}] (* Harvey P. Dale, Jul 16 2025 *)

a(n) = prime(n^2-1) = prime(A000290(n) - 1) = prime(A005563(n-1)).

A222760 Conjectured least prime p for which binomial(n*q,q) (mod q^3) = n for all primes q >= p.

Original entry on oeis.org

2, 5, 3, 2, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

T. D. Noe, Mar 13 2013

nonn

The n = 2 case is mentioned in Eric Weisstein's website.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric W. Weisstein, MathWorld: Central Binomial Coefficient

Cf. A096328 (prime(prime(n)^3)), A222759.

lim = 100; Table[r = Table[Mod[Binomial[n*p, p], p^3] == n, {p, Prime[Range[lim]]}]; i = lim; While[i > 0 && r[[i]], i--]; Prime[i + 1], {n, 87}]

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

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A243896 a(n) = prime(n^2+1).

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A243895 a(n) = prime(n^2-1).

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A222760 Conjectured least prime p for which binomial(n*q,q) (mod q^3) = n for all primes q >= p.

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