A256613 Numbers D such that D^2 = A^3 + B^4 + C^5 and A^2 + B^3 + C^4 = d^2 for some positive integers A, B, C, d.

7, 9, 17, 55, 72, 96, 459, 616, 1536, 4125, 9504, 11875, 19551, 36864, 64881, 67392, 77824, 108000, 171699, 262656, 388869, 559776, 786375, 1052672, 1081344, 1160000, 1413872, 1459161, 1850202, 1936224, 2530971, 3264000, 4158189, 5187500, 5238816, 6533679

Offset: 1

Author

M. F. Hasler, Apr 04 2015

Keywords

• nonn

Comments

Subsequence of A256091 such that A^2 + B^3 + C^4 = A180241(k)^2 for some k.

For A=2^(2n+2), B=2^(2n+1), C=2^(2n), n=0,1,2,... one has A^2+B^3+C^4 = 2^(4n) (16+8*4^n+16^n) = d^2 with d = 4^n (4^n+4), and A^3+B^4+C^5 = 2^(6n) (2^(2n)+8)^2 = D^2 with D = 8^n (4^n+8). So the latter represents an infinite subsequence (9, 96, 1536, ...) of this sequence.

Examples

			(A, B, C) = (1, 4, 2) = 1^3 + 4^4 + 2^5 = 1 + 256 + 32 = 289 = 17^2
and 1^2 + 4^3 + 2^4 = 1 + 64 + 16 = 81 = 9^2,
so 17 is a term.
		

Crossrefs

Cf. A256091, A180241, A180242, A255830.

Programs

PARI

is_A256613(D)={my(A,C=0,D2C5);while(1A256613(D) && print1(D","))}

Extensions

Inserted a(3)=17 and added a(18-36) by Lars Blomberg, Apr 26 2015