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A349420 Primes that do not divide any term of A275027.

%I A349420 #20 May 31 2024 21:01:45
%S A349420 2,3,7,11,31,41,67,73,79,89,97,101,103,107,127,131,137,181,211,251,
%T A349420 277,281,283,293,307,311,317,331,347,349,359,367,383,409,419,421,431,
%U A349420 449,463,523,547,563,577,599,607,613,617,631,677,683,691,773,787,797,821,823,827,911,977
%N A349420 Primes that do not divide any term of A275027.
%C A349420 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.
%H A349420 Michel Marcus, <a href="/A349420/b349420.txt">Table of n, a(n) for n = 1..289</a>
%H A349420 Joel A. Henningsen and Armin Straub, <a href="https://arxiv.org/abs/2111.08641">Generalized Lucas congruences and linear p-schemes</a>, arXiv:2111.08641 [math.NT], 2021.
%t A349420 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 *)
%o A349420 (PARI) f(n) = sum(k=0, n, binomial(n, k)^2*binomial(n-k, k)); \\ A275027
%o A349420 isdiv(v, n) = {my(p=prime(n)); for (k=1, p, if (!(v[k] % p), return(1));); return(0);}
%o A349420 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);}
%Y A349420 Cf. A275027.
%K A349420 nonn
%O A349420 1,1
%A A349420 _Michel Marcus_, Nov 17 2021