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
KEYWORD
nonn
AUTHOR
Michel Marcus, Nov 17 2021
STATUS
approved