%I #21 Aug 23 2018 17:05:18

%S 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,

%T 125,126,128,135,189,225,245,250,256,343,375,625,1029,2401,4375

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

%C 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.

%C 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.

%C Among them, exactly 63 solutions consist of 7-smooth numbers, which yields exactly 40 terms of this sequence.

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

%H B. M. M. de Weger, <a href="https://www.win.tue.nl/~bdeweger/proefschrift.html">Algorithms for Diophantine Equations</a>, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.

%e a(13) = 16 = 1 + 15 = 7 + 9 = 2^4 = 1 + 3 * 5 = 7 + 3^2.

%e a(25) = 81 = 1 + 80 = 25 + 56 = 32 + 49 = 3^4 = 1 + 2^4 * 5 = 5^2 + 2^3 * 7 = 2^5 + 7^2.

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

%Y Cf. A085153 (subsequence)

%K nonn,fini,full

%O 1,1

%A _Tomohiro Yamada_, May 29 2018