A274219 Numbers n such that n^k is the average of a positive square and a positive cube for all k > 0.

1, 64, 729, 1000, 1728, 4096, 13456, 15625, 27000, 46656, 59319, 64000, 110592, 117649, 148877, 262144, 531441, 729000, 777924, 861184, 1000000, 1259712, 1537600, 1728000

Offset: 1

Author

Altug Alkan, Jun 14 2016

Keywords

• nonn
• more

Comments

Terms of this sequence are 1, 2^6, 3^6, 10^3, 12^3, 2^12, 116^2, 5^6, 30^3, ...

Examples

			13456 = 116^2 is a term because 13456 = (163^2 + 7^3) / 2, 13456^2 = (1088^2 + 712^3) / 2, 13456^3 = ((116^3)^2 + (116^2)^3) / 2, 13456^4 = (254426048^2 + 94192^3) / 2, 13456^5 = (1698254848^2 + 9580672^3) / 2, 13456^6 = ((116^6)^2 + (116^4)^3) / 2.
		

Crossrefs

Cf. A055394.

Programs

PARI

isA055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
isok(n) = isA055394(2*n) && isA055394(2*n^2) && isA055394(2*n^3) && isA055394(2*n^4) && isA055394(2*n^5)

Extensions

a(13)-a(18) from Giovanni Resta, Jun 14 2016