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A303332 7-smooth numbers representable as the sum of two relatively prime 7-smooth 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 (list; graph; refs; listen; history; text; internal format)
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 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.

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

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Last modified August 14 13:34 EDT 2020. Contains 336480 sequences. (Running on oeis4.)