A349420 Primes that do not divide any term of A275027.

2, 3, 7, 11, 31, 41, 67, 73, 79, 89, 97, 101, 103, 107, 127, 131, 137, 181, 211, 251, 277, 281, 283, 293, 307, 311, 317, 331, 347, 349, 359, 367, 383, 409, 419, 421, 431, 449, 463, 523, 547, 563, 577, 599, 607, 613, 617, 631, 677, 683, 691, 773, 787, 797, 821, 823, 827, 911, 977

Offset: 1

Author

Michel Marcus, Nov 17 2021

Keywords

• nonn

Comments

f(n) = A275027(n) is never divisible by a prime p if none of the values f(0), f(1), ..., f(p-1) is divisible by p. See Henningsen and Straub, who ask for an explicit characterization for these primes.

Links

• Michel Marcus, Table of n, a(n) for n = 1..289
• Joel A. Henningsen and Armin Straub, Generalized Lucas congruences and linear p-schemes, arXiv:2111.08641 [math.NT], 2021.

Crossrefs

Cf. A275027.

Programs

Mathematica

f[n_] := f[n] = Sum[Binomial[n, k]^2*Binomial[n - k, k], {k, 0, n/2}]; q[p_] := AllTrue[Table[f[k], {k, 2, p - 1}], ! Divisible[#, p] &]; Select[Range[1000], PrimeQ[#] && q[#] &] (* Amiram Eldar, Nov 17 2021 *)

PARI

f(n) = sum(k=0, n, binomial(n, k)^2*binomial(n-k, k)); \\ A275027
isdiv(v, n) = {my(p=prime(n)); for (k=1, p, if (!(v[k] % p), return(1));); return(0);}
lista(nn) = {my(p=prime(nn), v=vector(p, k, f(k-1)), list=List()); for(n=1, nn, if (! isdiv(v, n), listput(list, prime(n)););); Vec(list);}