A376925 a(n) is the largest number that can be written as x + y with x and y coprime and such that each of x, y, and x + y are prime(n)-smooth.

2, 9, 128, 4375, 18225, 1771561, 27295744, 1375325568, 6313843404, 289478389760

Offset: 1

Author

Zhicheng Wei, Oct 10 2024

Keywords

• nonn
• more

Comments

The terms given here can be verified by checking that the number of solutions up to a(n) equals A362593(n) and a(n) is a solution x+y.

a(2) = 9 corresponds to Catalan's conjecture (Mihăilescu's theorem).

a(4) = 4375 corresponds to the final term of A303332.

Examples

			a(3) = 128 because prime(3) = 5, and 125 + 3 = 128 with 125 and 3 coprime, and 125, 3 and 128 are all 5-smooth numbers, and no number larger than 128 has these properties.
Table x + y = a(n) is shown below (q gives abc triple quality):
  n=1: 1 + 1 = 2 (q=1),
  n=2: 8 + 1 = 9 (q=1.226)
  n=3: 125 + 3 = 128 (q=1.426)
  n=4: 4374 + 1 = 4375 (q=1.567)
  n=5: 14641 + 3584 = 18225 (q=1.267)
  n=6: 1771470 + 91 = 1771561 (q=1.395)
  n=7: 27217619 + 78125 = 27295744 (q=1.421)
  n=8: 1371299293 + 4026275 = 1375325568 (q=1.31)
  n=9: 4867359029 + 1446484375 = 6313843404 (q=1.17)
  n=10: 289478257991 + 131769  = 289478389760 (q=1.16)
		

Links

• Wikipedia, Abc conjecture.
• Wikipedia, Catalan's conjecture.

Crossrefs

Cf. A303332, A362593.

Extensions

a(7)-a(8) from Andrew Howroyd, Oct 12 2024

a(9)-a(10) from David A. Corneth, Nov 24 2024