cp's OEIS Frontend

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.

%I A256613 #22 May 01 2015 09:14:56
%S A256613 7,9,17,55,72,96,459,616,1536,4125,9504,11875,19551,36864,64881,67392,
%T A256613 77824,108000,171699,262656,388869,559776,786375,1052672,1081344,
%U A256613 1160000,1413872,1459161,1850202,1936224,2530971,3264000,4158189,5187500,5238816,6533679
%N 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.
%C A256613 Subsequence of A256091 such that A^2 + B^3 + C^4 = A180241(k)^2 for some k.
%C A256613 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.
%e A256613 (A, B, C) = (1, 4, 2) = 1^3 + 4^4 + 2^5 = 1 + 256 + 32 = 289 = 17^2
%e A256613 and 1^2 + 4^3 + 2^4 = 1 + 64 + 16 = 81 = 9^2,
%e A256613 so 17 is a term.
%o A256613 (PARI) is_A256613(D)={my(A,C=0,D2C5);while(1<D2C5=D^2-C++^5,for(B=1,sqrtint(sqrtint(D2C5-1)), ispower(D2C5-B^4,3,&A) && issquare(C^4+B^3+A^2) && return(1)))}
%o A256613 {for(D=3,10^5,is_A256613(D) && print1(D","))}
%Y A256613 Cf. A256091, A180241, A180242, A255830.
%K A256613 nonn
%O A256613 1,1
%A A256613 _M. F. Hasler_, Apr 04 2015
%E A256613 Inserted a(3)=17 and added a(18-36) by _Lars Blomberg_, Apr 26 2015