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 A300332 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Peter Luschny, Table of n, a(n) for n = 1..10000 Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017. EXAMPLE 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), (p^p-1)/(p-1),   (A023037), p^p,             (A000312, A051674). . 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. PROG (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)) CROSSREFS Indices of the nonzero values of A300333. Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645. Sequence in context: A164831 A085188 A286728 * A244819 A305185 A083561 Adjacent sequences:  A300329 A300330 A300331 * A300333 A300334 A300335 KEYWORD nonn AUTHOR Peter Luschny, Mar 03 2018 STATUS approved

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Last modified October 1 17:45 EDT 2020. Contains 337444 sequences. (Running on oeis4.)