A373747 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).

1, 9, 21, 185, 297, 341, 405, 861, 1113, 1645, 1677, 1833, 2409, 3417, 3621, 4545, 6141, 8549, 8949, 8961, 9309, 10205, 11049, 12441, 15621, 16617, 17313, 18093, 18357, 19401, 19749, 20241, 20793, 21605, 21645, 21837, 22017, 22765, 24753, 25197, 25573, 26469

Offset

1,2

Comments

The congruence holds for all prime numbers p such that p == 1 (mod 4) (Sawhney, 2017).

Links

Table of n, a(n) for n=1..42.

Mehtaab Sawhney, Problem H-815, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 55, No. 4 (2017), p. 374; A congruence with powers of 2 and binomial coefficients, Solution to Problem H-815 by the proposer, ibid., Vol. 57, No. 4 (2019), pp. 377-378.

Example

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).

Mathematica

q[n_] := Divisible[Sum[2^k*Binomial[3*k, k], {k, 0, n - 1}] - 1, n]; Select[4*Range[0, 250] + 1, ! PrimeQ[#] && q[#] &]

Prog

(PARI) is(k) = (k % 4 == 1) && !isprime(k) && sum(j = 0, k-1, Mod(2, k)^j * binomial(3*j, j)) == 1;

Crossrefs

Cf. A002144, A005809, A091113.

Sequence in context: A050860 A339724 A342409 * A329007 A368120 A250729

Adjacent sequences: A373744 A373745 A373746 * A373748 A373749 A373750

Keyword

nonn

Author

Amiram Eldar, Jun 18 2024

Status

approved