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A300332
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Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.
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4
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3, 4, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 80, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199
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OFFSET
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1,1
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COMMENTS
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Equivalently these are the integers represented by a cyclotomic binary form Phi_p(x,y) where p is prime and x and y are positive integers with max(x,y) >= 2. A cyclotomic binary form (over Z) is a homogeneous polynomial in two variables of the form f(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function.
An efficient and safe calculation of this sequence requires a precise knowledge of the range of possible solutions of the associated Diophantine equations. The bounds used in the Julia program below were specified by Fouvry, Levesque and Waldschmidt.
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LINKS
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EXAMPLE
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Let p denote an odd prime. Subsequences are numbers of the form
2^p - 1, (A001348) (x = 1, y = 2) (Mersenne numbers),
p*2^(p - 1), (A299795) (x = 2, y = 2),
(3^p - 1)/2, (A003462) (x = 1, y = 3),
3^p - 2^p, (A135171) (x = 2, y = 3),
p*3^(p - 1), (A027471) (x = 3, y = 3),
(4^p - 1)/3, (A002450) (x = 1, y = 4),
2^(p-1)*(2^p-1), (A006516) (x = 2, y = 4),
4^p - 3^p, (A005061) (x = 3, y = 4),
p*4^(p - 1), (A002697) (x = 4, y = 4),
.
The generalized cuban primes A007645 are a subsequence, as are the quintan primes A002649, the septan primes and so on.
All primes in this sequence less than 1031 are generalized cuban primes. 1031 is an element because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, however 1031 is not a cuban prime because 1030 is not divisible by 6.
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PROG
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(Julia)
using Primes
function isA300332(n)
logn = log(n)^1.161
K = Int(floor(5.383*logn))
M = Int(floor(2*(n/3)^(1/2)))
k = 2
while k <= K
if k == 7
K = Int(floor(4.864*logn))
M = Int(ceil(2*(n/11)^(1/4)))
end
for y in 2:M, x in 1:y
r = x == y ? k*y^(k - 1) : div(x^k - y^k, x - y)
n == r && return true
end
k = nextprime(k+1)
end
return false
end
A300332list(upto) = [n for n in 1:upto if isA300332(n)]
println(A300332list(200))
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CROSSREFS
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Indices of the nonzero values of A300333.
Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645.
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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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