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A274033
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Numbers n such that n = a^2 + b^4 and n^2 = c^3 + d^5 for some positive integers a, b, c, d.
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0
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81250, 1062882, 11529602, 12500000, 170061120, 200000000, 2662400000, 5897400777, 7309688832, 12814453125, 34297420960, 37019531250
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..12.
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EXAMPLE
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81250 is a term because 81250 = 175^2 + 15^4 and 81250^2 = 1875^3 + 25^5.
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CROSSREFS
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Cf. A100293, A111925.
Sequence in context: A252625 A233994 A237942 * A029752 A043608 A249231
Adjacent sequences: A274030 A274031 A274032 * A274034 A274035 A274036
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KEYWORD
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nonn,more
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AUTHOR
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Altug Alkan, Jun 07 2016
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EXTENSIONS
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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
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STATUS
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approved
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