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A321671
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Primes of the form 2^j - 3^k, for j >= 0, k >= 0.
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4
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3, 5, 7, 13, 23, 29, 31, 37, 47, 61, 101, 127, 229, 269, 431, 503, 509, 997, 1021, 1319, 2039, 3853, 4093, 7949, 8111, 8191, 14197, 16141, 16381, 32687, 45853, 65293, 130343, 130829, 131063, 131071, 347141, 502829, 524261, 524287, 1028893, 1046389, 1048549
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OFFSET
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1,1
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COMMENTS
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The numbers in A007643 are not in this sequence.
For n > 1, a(n) is of the form 8k - 1 or 8k - 3.
In this sequence, only 3 and 7 make both j and k even numbers.
Generally, the way to prove that a number is not in this sequence is to successively take residues modulo 3, 8, 5, and 16 on both sides of the equation 2^j - 3^k = x.
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LINKS
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H. Gauchman and I. Rosenholtz (Proposers), R. Martin (Solver), Difference of prime powers, Problem 1404, Math. Mag., 65 (No. 4, 1992), 265; Solution, Math. Mag., 66 (No. 4, 1993), 269.
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FORMULA
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EXAMPLE
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7 = 2^3 - 3^0, so 7 is a term.
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PROG
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(PARI) forprime(p=1, 1000, k=0; x=2; y=1; while(k<p+1, while(x<y+p, x=2*x); if(x-y==p, print1(p, ", "); k=p); k++; y=3*y))
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CROSSREFS
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Cf. A004051 (primes of the form 2^a + 3^b).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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