A373734 - OEIS

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A373734

Odd composite numbers k such that Sum_{i=1..k-1} 2^i * i^(k-2) == Sum_{i=1..(k-1)/2} i^(k-2) (mod k).

49, 111, 343, 561, 637, 905, 1105, 1519, 1729, 2465, 2613, 2821, 4017, 6517, 6601, 8029, 8911, 10585, 10621, 11973, 15841, 20091, 20301, 29341, 41041, 46657, 47677, 52633, 54145, 62745, 63973, 75361, 94285, 99269, 101101, 101185, 115921, 126217, 134113, 138229

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OFFSET

1,1

COMMENTS

The congruence holds for all odd primes (A065091).

Apparently, all the Carmichael numbers (A002997) are terms (checked for the first 500 Carmichael numbers).

LINKS

Table of n, a(n) for n=1..40.

Christopher J. Hillar, Problem 10723, The American Mathematical Monthly, Vol. 106, No. 3 (1999), p. 265; Two Sums That Are Congruent Modulo p, solution to Problem 10723 by Heinz-Jiirgen Seiffert, ibid., Vol. 108, No. 2 (2001), p. 176.

MATHEMATICA

f[p_] := Sum[PowerMod[2, i, p]*PowerMod[i, p - 2, p], {i, 1, p - 1}] - Sum[PowerMod[i, p - 2, p], {i, 1, (p - 1)/2}]; q[p_] := CompositeQ[p] && Divisible[f[p], p]; Select[Range[1, 10000, 2], q]

PROG

(PARI) is(k) = (k > 1) && (k % 2) && !isprime(k) && sum(i = 1, k-1, Mod(2, k)^i * Mod(i, k)^(k-2)) == sum(i = 1, (k-1)/2, Mod(i, k)^(k-2));

CROSSREFS

Cf. A002997, A065091.

Sequence in context: A250653 A045253 A088868 * A044236 A044617 A216169

Adjacent sequences: A373731 A373732 A373733 * A373735 A373736 A373737

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jun 18 2024

STATUS

approved