

A274033


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


0



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

1,1


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.


LINKS

Table of n, a(n) for n=1..12.


EXAMPLE

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


CROSSREFS

Cf. A100293, A111925.
Sequence in context: A252625 A233994 A237942 * A029752 A043608 A249231
Adjacent sequences: A274030 A274031 A274032 * A274034 A274035 A274036


KEYWORD

nonn,more


AUTHOR

Altug Alkan, Jun 07 2016


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


STATUS

approved



