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A303332
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7-smooth numbers representable as the sum of two relatively prime 7-smooth numbers.
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0
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2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 21, 25, 27, 28, 32, 35, 36, 49, 50, 54, 64, 81, 125, 126, 128, 135, 189, 225, 245, 250, 256, 343, 375, 625, 1029, 2401, 4375
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OFFSET
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1,1
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COMMENTS
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It follows from Theorem 6.3 of de Weger's tract that there are exactly 40 terms, the largest of which is 4375 = 1 + 4374 = 5^4 * 7 = 1 + 2 * 3^7.
Indeed, de Weger determined all solutions of the equation x + y = z with x, y, z 13-smooth, x, y relatively prime and x <= y; there exist exactly 545 solutions.
Among them, exactly 63 solutions consist of 7-smooth numbers, which yields exactly 40 terms of this sequence.
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REFERENCES
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T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.
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LINKS
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EXAMPLE
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a(13) = 16 = 1 + 15 = 7 + 9 = 2^4 = 1 + 3 * 5 = 7 + 3^2.
a(25) = 81 = 1 + 80 = 25 + 56 = 32 + 49 = 3^4 = 1 + 2^4 * 5 = 5^2 + 2^3 * 7 = 2^5 + 7^2.
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MATHEMATICA
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s7 = Select[Range[10000], FactorInteger[#][[-1, 1]] <= 7 &]; Select[s7, AnyTrue[ IntegerPartitions[#, {2}, s7], GCD @@ # == 1 &] &] (* Giovanni Resta, May 30 2018 *)
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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