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A373763
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Numbers k that are neither primes nor squares of primes such that A006134(k) - A102283(k) is divisible by k.
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1
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1, 27, 63, 81, 238, 243, 247, 279, 322, 580, 671, 729, 1222, 2074, 2187, 3172, 3550, 3577, 4185, 5589, 6561, 7805, 7957, 8239, 8701, 8890, 9040, 9064, 9523, 9730, 9898, 10087, 10138, 10549, 11074, 11176, 11440, 11473, 11920, 12232, 12430, 12604, 13900, 14287, 14410
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OFFSET
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1,2
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COMMENTS
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The congruence A006134(k) == A102283(k) (mod k) holds for all values of k that are primes or squares of primes.
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LINKS
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MATHEMATICA
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q[n_] := !PrimeQ[n] && !PrimeQ[Sqrt[n]] && Divisible[Sum[Binomial[2*k, k], {k, 0, n - 1}] - JacobiSymbol[n, 3], n]; Select[Range[1000], q]
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PROG
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(PARI) is1(k) = !isprime(k) && !(issquare(k) && isprime(sqrtint(k)));
lista(kmax) = {my(s0 = 1, s1 = 3); print1(1, ", "); for(k = 2, kmax, s2 = ((5*k - 2) * s1 - 2 * (2*k - 1) * s0 )/k; if(is1(k + 1) && !((s2 - [1, -1, 0][k % 3 + 1]) % (k + 1)), print1(k + 1, ", ")); s0 = s1; s1 = s2); }
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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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