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

1,1

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.

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 *)

Prog

(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); }

Crossrefs

Cf. A275027.

Sequence in context: A120856 A138000 A322118 * A323067 A140108 A034295

Adjacent sequences: A349417 A349418 A349419 * A349421 A349422 A349423

Keyword

nonn,changed

Author

Michel Marcus, Nov 17 2021

Status

approved