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A256613
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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.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..36.
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EXAMPLE
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(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.
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PROG
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(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)))}
{for(D=3, 10^5, is_A256613(D) && print1(D", "))}
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CROSSREFS
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Cf. A256091, A180241, A180242, A255830.
Sequence in context: A194830 A183344 A255830 * A116484 A138749 A320700
Adjacent sequences: A256610 A256611 A256612 * A256614 A256615 A256616
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KEYWORD
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nonn
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AUTHOR
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M. F. Hasler, Apr 04 2015
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EXTENSIONS
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Inserted a(3)=17 and added a(18-36) by Lars Blomberg, Apr 26 2015
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STATUS
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approved
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