Search: a002144 -id:a002144
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A373747
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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).
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+10
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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
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OFFSET
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1,2
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COMMENTS
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The congruence holds for all prime numbers p such that p == 1 (mod 4) (Sawhney, 2017).
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LINKS
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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.
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EXAMPLE
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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).
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MATHEMATICA
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q[n_] := Divisible[Sum[2^k*Binomial[3*k, k], {k, 0, n - 1}] - 1, n]; Select[4*Range[0, 250] + 1, ! PrimeQ[#] && q[#] &]
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PROG
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(PARI) is(k) = (k % 4 == 1) && !isprime(k) && sum(j = 0, k-1, Mod(2, k)^j * binomial(3*j, j)) == 1;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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