A274033 Numbers k such that k = a^2 + b^4 and n^2 = c^3 + d^5 for some positive integers a, b, c, d.

81250, 1062882, 11529602, 12500000, 170061120, 200000000, 2662400000, 5897400777, 7309688832, 12814453125, 34297420960, 37019531250

Offset: 1

Author

Altug Alkan, Jun 07 2016

Keywords

• nonn
• more

Comments

In other words, values of a^2 + b^4 such that (a^2 + b^4)^2 is of the form c^3 + d^5 where a, b, c, d > 0.

81250 is the least number with this property.

Sequence is infinite: If n = a^2 + b^4 and n^2 = c^3 + d^5, then n*k^60 = (a*k^30)^2 + (b*k^15)^4 and (n*k^60)^2 = (c*k^40)^3 + (d*k^24)^5. So if n is in this sequence, then n*k^60 is in this sequence for all nonzero values of k.

Examples

			81250 is a term because 81250 = 175^2 + 15^4 and 81250^2 = 1875^3 + 25^5.
		

Crossrefs

Cf. A100293, A111925.

Extensions

a(2)-a(6) from Giovanni Resta, Jun 07 2016

a(7) from Chai Wah Wu, Jun 14 2016

a(8)-a(12) from Chai Wah Wu, Jul 07 2016