Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 Jun 22 2024 22:41:03
%S 1,9,21,185,297,341,405,861,1113,1645,1677,1833,2409,3417,3621,4545,
%T 6141,8549,8949,8961,9309,10205,11049,12441,15621,16617,17313,18093,
%U 18357,19401,19749,20241,20793,21605,21645,21837,22017,22765,24753,25197,25573,26469
%N Nonprime numbers k of the form 4*m+1 such that Sum_{j=0..k-1} 2^j * binomial(3*j, j) == 1 (mod k).
%C The congruence holds for all prime numbers p such that p == 1 (mod 4) (Sawhney, 2017).
%H Mehtaab Sawhney, <a href="https://www.fq.math.ca/Problems/November2017advanced.pdf">Problem H-815</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 55, No. 4 (2017), p. 374; <a href="https://www.fq.math.ca/Problems/Nov2019AdvProb.pdf">A congruence with powers of 2 and binomial coefficients</a>, Solution to Problem H-815 by the proposer, ibid., Vol. 57, No. 4 (2019), pp. 377-378.
%e 9 is a term since 9 = 3^3 is nonprime, 9 = 4*2 + 1, and Sum_{j=0..8} 2^j * binomial(3*j,j) = 204457267 == 1 (mod 9).
%t q[n_] := Divisible[Sum[2^k*Binomial[3*k, k], {k, 0, n - 1}] - 1, n]; Select[4*Range[0, 250] + 1, ! PrimeQ[#] && q[#] &]
%o (PARI) is(k) = (k % 4 == 1) && !isprime(k) && sum(j = 0, k-1, Mod(2, k)^j * binomial(3*j, j)) == 1;
%Y Cf. A002144, A005809, A091113.
%K nonn
%O 1,2
%A _Amiram Eldar_, Jun 18 2024