A256603 -id:A256603 - cp's OEIS frontend

A256604 Numbers D such that D^2 = A^2 + B^3 + C^4 has more than one solution in positive integers (A, B, C).

5, 9, 12, 17, 19, 21, 23, 25, 28, 33, 35, 37, 38, 39, 42, 45, 46, 47, 51, 53, 55, 57, 59, 60, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 77, 80, 81, 82, 84, 87, 88, 89, 91, 93, 95, 97, 98, 99, 100, 103, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 123, 124, 127, 128, 129, 131, 132, 133, 134, 135, 136, 139, 141

Offset: 1

M. F. Hasler, Apr 06 2015

nonn

The subsequence of terms of A180241 whose square has more than one representation of the given form. See A256603 and A256652 are the analog for A256091 and A255830.

			(A, B, C) = (4, 8, 1): 4^2 + 8^3 + 1^4 = 16 + 512 + 1 = 529 = 23^2 and
(A, B, C) = (1, 8, 2): 1^2 + 8^3 + 2^4 = 1 + 512 + 16 = 529 = 23^2,
so 23 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..10088

Cf. A180241, A180242, A256613, A255830.

for(D=2,199,my(f=-1,B,D2C4);for(C=1,sqrtint(D),D2C4=D^2-C^4; B=0;while(B++^3M. F. Hasler, May 01 2015

Inserted a(7)=23 by Lars Blomberg, Apr 26 2015

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

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Apr 06 2015

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

			(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.

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

is_A256652(D,f=-1)={my(C=0,B,D2C6);while(1A256652(D)&&print1(D",")) \\ Converted to integer arithmetic by M. F. Hasler, May 01 2015

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

A257298 Numbers whose cube is of the form a^5 + b^5 - c^5 with a >= b > 0 and c not in {a,b}.

Original entry on oeis.org

144, 3969, 4114, 4608, 17918, 18723, 34992, 44944, 53176, 75076, 127008, 131648, 147456, 163500, 171698, 206116, 235225, 347778, 450000, 462220, 573376, 599136, 611524, 660969, 715716, 927799, 943020, 964467, 986049, 999702

Offset: 1

M. F. Hasler, May 21 2015

nonn

Otherwise said, d-values of nontrivial solutions to a^5 + b^5 = c^5 + d^3.

			144^3 = 192^5 + 156^5 - 204^5,
3969^3 = 126^5 + 126^5 - 63^5,
4114^3 = 143^5 + 121^5 - 110^5,
4608^3 = 1536^5 + 1248^5 - 1632^5, ...

Cf. A000578, A000584, A020896, A256603, A255351.

Most terms computed by Giovanni Resta, May 25 2015

A266967 Numbers D such that D^2 = A^3 + B^4 + C^5 has more than two solutions in positive integers (A, B, C).

Original entry on oeis.org

77824, 1052672, 5921875

Offset: 1

Chai Wah Wu, Jan 07 2016

Subsequence of A256603.

			640^3 + 224^4 + 80^5 = 1536^3 + 192^4 + 64^5 = 1792^3 + 128^4 + 32^5 = 6056574976 = 77824^2.
768^3 + 1024^4 + 96^5 = 2048^3 + 64^4 + 256^5 = 10240^3 + 64^4 + 128^5 = 1108118339584 = 1052672^2.
6250^3 + 1375^4 + 500^5 = 15625^3 + 250^4 + 500^5 = 31250^3 + 1375^4 + 250^5 = 35068603515625 =  5921875^2.

Cf. A256603, A256091

cp's OEIS Frontend

A256604 Numbers D such that D^2 = A^2 + B^3 + C^4 has more than one solution in positive integers (A, B, C).

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

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A257298 Numbers whose cube is of the form a^5 + b^5 - c^5 with a >= b > 0 and c not in {a,b}.

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A266967 Numbers D such that D^2 = A^3 + B^4 + C^5 has more than two solutions in positive integers (A, B, C).

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