A256604 -id:A256604 - cp's OEIS frontend

A256603 Numbers D such that D^2 = A^3 + B^4 + C^5 has more than one solution in positive integers (A, B, C).

305, 525, 1206, 1257, 1395, 2048, 2213, 3072, 4348, 6400, 16385, 16640, 16704, 20631, 22872, 23256, 30968, 31407, 32769, 62943, 74515, 77713, 77824, 79776, 82565, 84775, 90432, 98739, 117600, 121250, 133696, 163525, 165628, 171576, 198400, 199872, 243225

Offset: 1

M. F. Hasler, Apr 06 2015

nonn

A subsequence of A256091. Sequences A256604 and A256652 are the analog for A180241 and A255830.

			(A, B, C) = (32, 128, 1): 32^3 + 128^4 + 1^5 = 32768 + 268435456 + 1 = 268468225 = 16385^2
(A, B, C) = (1, 128, 8): 1^3 + 128^4 + 8^5 = 1 + 268435456 + 32768 = 268468225 = 16385^2
so 16385 is a term.

Chai Wah Wu, Table of n, a(n) for n = 1..185

Cf. A180241, A180242, A255830, A256091, A256604, A256613, A256652.

for(D=1,9999,f=-1;for(C=1,sqrtn(D^2-1,5),for(B=1,sqrtn(D^2-C^5-.5,4),ispower(D^2-C^5-B^4,3)&&f++&print1(D",")+next(3))))

Inserted a(11),a(16) and added a(19)-a(37) by Lars Blomberg, Apr 17 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

A267216 Numbers D such that D^2 = A^2 + B^3 + C^4 has more than two solutions in positive integers (A, B, C).

Original entry on oeis.org

9, 21, 28, 33, 45, 47, 51, 53, 55, 61, 65, 66, 68, 69, 70, 73, 75, 77, 81, 82, 84, 87, 89, 91, 93, 95, 97, 103, 105, 107, 108, 109, 110, 111, 113, 114, 116, 117, 119, 123, 128, 129, 131, 133, 135, 136, 139, 142, 143, 145, 147, 149, 150, 152, 154, 156, 157, 161

Offset: 1

Chai Wah Wu, Feb 01 2016

nonn

Subsequence of A256604.

			9^2 = 1^2 + 4^3 + 2^4 = 4^2 + 4^3 + 1^4 = 8^2 + 1^3 + 2^4.
21^2 = 11^2 + 4^3 + 4^4 = 12^2 + 6^3 + 3^4 = 19^2 + 4^3 + 2^4.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A256604.

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A256603 Numbers D such that D^2 = A^3 + B^4 + C^5 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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A267216 Numbers D such that D^2 = A^2 + B^3 + C^4 has more than two solutions in positive integers (A, B, C).

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