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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).
0
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
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
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
Sequence in context: A250653 A045253 A088868 * A044236 A044617 A216169
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 18 2024
STATUS
approved