A256652 Numbers D such that D^2 = A^4 + B^5 + C^6 has more than one solution in positive integers (A, B, C).

1257, 32769, 262176, 262208, 1081344, 4198400, 16777217, 16809984

Offset

1,1

Comments

A subsequence of A255830. Sequences A256604 and A256603 are the analog for A180241 and A256091.

Terms a(2) - a(8) have Hamming weight 2: 32769 = 2^15 + 1, 262176 = 2^18 + 2^5, 262208 = 2^18 + 2^6, 1081344 = 2^20 + 2^15, 4198400 = 2^22 + 2^12, 16777217 = 2^24 + 1, 16809984 = 2^24 + 2^15.

Given D^2 = A^4+B^5+C^6, multiply by u^60, u>1, to get (u^30*D)^2 = (u^15*A)^4 + (u^12*B)^5 + (u^10*C)^6. If D is a solution then so is u^30*D. - Lars Blomberg, Apr 26 2015

Solutions for a(1)-a(8) as well as some larger terms:

..A1.....B1....C1......A2.....B2....C2..............D

..35......8.....6......32......2.....9...........1257

..16......1....32......16.....64.....1..........32769

..64......4....64.....512......4....16.........262176

...8.....32....64.....512.....32.....4.........262208

1024.....64....64.....512....256....32........1081344

.480....240...160....2048....128....16........4198400

...1.....32...256....4096.....32.....1.......16777217

1024.....64...256....4096....256....32.......16809984

.512......4..1024...32768......4....64.....1073741856

1024...4096.....8...32768....256.....8.....1073742336

4096...2048..1024...32768...2048...256.....1090519040

...1..16384....64.....512..16384.....1....34359738369

4096..16384....16......64..16384...256....34359742464

4096..16384..1024...32768..16384...256....34376515584

.512...2048..4096..262144...2048....64....68719738880

...1....256..8192....1024......1..8192...549755813889

1024...4096..8192...32768....256..8192...549756862464

- Lars Blomberg, Apr 26 2015

Links

Table of n, a(n) for n=1..8.

Example

(A, B, C) = (32, 2, 9): 32^4 + 2^5 + 9^6 = 1048576 + 32 + 531441 = 1580049 = 1257^2, and
(A, B, C) = (35, 8, 6): 35^4 + 8^5 + 6^6 = 1500625 + 32768 + 46656 = 1580049 = 1257^2,
so 1257 is a term.

Prog

(PARI) is_A256652(D, f=-1)={my(C=0, B, D2C6); while(1<D2C6=D^2-C++^6, B=0; while(0<D2C6-B++^5, ispower(D2C6-B^5, 4)&&f++&&return(1)))}
for(D=2, 10^5, is_A256652(D)&&print1(D", ")) \\ Converted to integer arithmetic by M. F. Hasler, May 01 2015

Crossrefs

Cf. A255830, A256603, A256091, A256613, A256604, A180241, A180242.

Sequence in context: A216941 A159726 A048130 * A211773 A215991 A288921

Adjacent sequences: A256649 A256650 A256651 * A256653 A256654 A256655

Keyword

nonn,bref,more

Author

M. F. Hasler, Apr 06 2015

Extensions

a(5)-a(8) from Lars Blomberg, Apr 26 2015

Status

approved