

A303332


7smooth numbers representable as the sum of two relatively prime 7smooth numbers.


0



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

1,1


COMMENTS

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 13smooth, x, y relatively prime and x <= y; there exist exactly 545 solutions.
Among them, exactly 63 solutions consist of 7smooth numbers, which yields exactly 40 terms of this sequence.


REFERENCES

T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.


LINKS

Table of n, a(n) for n=1..40.
B. M. M. de Weger, Algorithms for Diophantine Equations, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.


EXAMPLE

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.


MATHEMATICA

s7 = Select[Range[10000], FactorInteger[#][[1, 1]] <= 7 &]; Select[s7, AnyTrue[ IntegerPartitions[#, {2}, s7], GCD @@ # == 1 &] &] (* Giovanni Resta, May 30 2018 *)


CROSSREFS

Cf. A085153 (subsequence)
Sequence in context: A096503 A055238 A100952 * A030477 A178859 A165412
Adjacent sequences: A303329 A303330 A303331 * A303333 A303334 A303335


KEYWORD

nonn,fini,full


AUTHOR

Tomohiro Yamada, May 29 2018


STATUS

approved



