A255830 -id:A255830 - cp's OEIS frontend

A180241 Numbers whose square can be expressed as the sum of a square, a cube and a fourth power, all positive.

5, 7, 8, 9, 12, 17, 18, 19, 21, 22, 23, 24, 25, 28, 31, 32, 33, 35, 37, 38, 39, 41, 42, 44, 45, 46, 47, 51, 52, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 93, 95, 96, 97, 98, 99, 100, 101

Offset: 1

Carmine Suriano, Aug 19 2010

nonn

a(n)^2 = x^2 + y^3 + z^4 with x,y,z > 0.

			a(5) = 12 is a term since 12*12 = 144 = 6^2 + 3^3 + 3^4 = 36 + 27 + 81.

Cf. A180242, A256091, A256613, A255830.

for(D=1,99,for(C=1,sqrtn(D^2-1,4),for(B=1,sqrtn(D^2-C^4-1,3),issquare(D^2-C^4-B^3,&A)&&print1([A,B,C,D][4]",")+next(3)))) \\ M. F. Hasler, Apr 06 2015

A180242 Numbers whose square cannot be expressed as the sum of a positive square, a positive cube and a positive fourth power.

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 11, 13, 14, 15, 16, 20, 26, 27, 29, 30, 34, 36, 40, 43, 48, 49, 50, 56, 58, 62, 64, 76, 86, 92, 94, 102, 104, 106, 122, 126, 130, 146, 148, 176, 178, 202, 211, 218, 227, 232, 238, 246, 248, 262, 272, 281, 286, 310, 311, 326, 335, 344, 346, 349, 370

Offset: 1

Carmine Suriano, Aug 19 2010

nonn

Complement to A180241 with respect to the set of positive integers.

If k^2 = m^2 + t^3 + u^4 where k, m, t and u are positive then k^2 - m^2 = (k - m)*(k + m) = t^3 + u^4 which might ease the search for terms by looking at divisors of t^3 + u^4. - David A. Corneth, Apr 03 2023

			a(5) = 6 since 6^2 = 36 cannot be expressed as the sum of a square, a cube and a fourth power.

David A. Corneth, Table of n, a(n) for n = 1..3170 (terms <= 10^7)

Cf. A180241, A255830, A256091, A256613.

for(D=1,99,for(C=1,sqrtn(D^2-1,4),for(B=1,sqrtn(D^2-C^4-1,3),issquare(D^2-C^4-B^3)&&next(3)));print1(D",")) \\ M. F. Hasler, Apr 06 2015

is(n)=my(n2=n^2); for(C=1, sqrtn(n2-1, 4), my(t=n2-C^4); for(B=1, sqrtn(t-1, 3), if(issquare(t-B^3), return(0)))); 1 \\ Charles R Greathouse IV, Apr 06 2015

Name clarified by David A. Corneth, Mar 20 2023

A256091 Numbers D such that D^2 = A^3 + B^4 + C^5 for some positive integers A, B, C.

Original entry on oeis.org

5, 7, 9, 11, 17, 23, 25, 33, 38, 45, 55, 72, 79, 89, 95, 96, 99, 100, 103, 105, 117, 133, 137, 163, 171, 213, 218, 220, 237, 239, 240, 248, 255, 257, 282, 303, 305, 320, 347, 355, 362, 375, 384, 393, 407, 408, 411, 459, 475, 503, 506, 513, 525, 539, 540, 567, 581, 613, 616, 657, 659, 660, 751, 752, 761, 792, 796, 808, 824, 833

Offset: 1

M. F. Hasler, Apr 04 2015

nonn

(8^n*(4^n+8); n = 0, 1, 2, ...) is an infinite subsequence of the subsequence A256613: see that entry for more details.

			(A, B, C) = (1, 4, 2): 1^3 + 4^4 + 2^5 = 1 + 256 + 32 = 289 = 17^2, so 17 is a term.

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

Cf. A180241, A180242, A256613, A255830.

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

for(D=3, 9999, ok = 0; for(C=1, sqrtn(D^2-2, 5), for(B=1, sqrtn(D^2-C^5-1, 4), ispower(D^2-C^5-B^4, 3)&&(ok=1)&&print1(D", "); if (ok, break)); if (ok, break))) \\ Michel Marcus, Apr 26 2015

Inserted a(5)=17 and removed the doublet 525 by Lars Blomberg, Apr 26 2015

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.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Apr 04 2015

nonn

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.

			(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.

Cf. A256091, A180241, A180242, A255830.

is_A256613(D)={my(A,C=0,D2C5);while(1A256613(D) && print1(D","))}

Inserted a(3)=17 and added a(18-36) by Lars Blomberg, Apr 26 2015

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A180241 Numbers whose square can be expressed as the sum of a square, a cube and a fourth power, all positive.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A180242 Numbers whose square cannot be expressed as the sum of a positive square, a positive cube and a positive fourth power.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Extensions

A256091 Numbers D such that D^2 = A^3 + B^4 + C^5 for some positive integers A, B, C.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions