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
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
KEYWORD
nonn,fini,full
AUTHOR
Tomohiro Yamada, May 29 2018
STATUS
approved