A180241 -id:A180241 - cp's OEIS frontend

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

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

A345645 Numbers whose square can be represented in exactly one way as the sum of a square and a biquadrate (fourth power).

Original entry on oeis.org

5, 15, 20, 34, 39, 41, 45, 60, 80, 85, 111, 125, 135, 136, 150, 156, 164, 175, 180, 194, 219, 240, 245, 255, 265, 306, 313, 320, 325, 340, 351, 353, 369, 371, 375, 405, 410, 444, 445, 455, 500, 505, 514, 540, 544, 600, 605, 609, 624, 629, 656, 671, 674, 689

Offset: 1

Mohammad Tejabwala, Jun 21 2021

nonn

Numbers z such that there is exactly one solution to z^2 = x^2 + y^4.
From Karl-Heinz Hofmann, Oct 21 2021: (Start)
No term can be a square (see the comment from Altug Alkan in A111925).
Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
Additionally, if the terms have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too. (End)

			3^2 + 2^4 = 9 + 16 = 25 = 5^2, so 5 is a term.
60^2 + 5^4 = 63^2 + 4^4 = 65^2, so 65 is not a term.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000290, A000583, A180241, A271576 (all solutions).
Cf. A345700 (2 solutions), A345968 (3 solutions), A346110 (4 solutions), A348655 (5 solutions), A349324 (6 solutions), A346115 (the least solutions).
Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).

Select[Range@100,Length@Solve[x^2+y^4==#^2&&x>0&&y>0,{x,y},Integers]==1&] (* Giorgos Kalogeropoulos, Jun 25 2021 *)

inlist(list, v) = for (i=1, #list, if (list[i]==v, return(1)));
isok(m) = {my(list = List()); for (k=1, sqrtnint(m^2, 4), if (issquare(j=m^2-k^4) && !inlist(vecsort([k^4,j^2])), listput(list, vecsort([k^4,j^2])));); #list == 1;} \\ Michel Marcus, Jun 26 2021

terms = []
for i in range(1, 700):
    occur = 0
    ii = i*i
    for j in range(1, i):
        k = int((ii - j*j) ** 0.25)
        if k*k*k*k + j*j == ii:
            occur += 1
    if occur == 1:
        terms.append(i)
print(terms)

A255830 Numbers D such that D^2 = A^4 + B^5 + C^6 for some positive integers A, B, C.

Original entry on oeis.org

7, 9, 17, 33, 72, 89, 96, 99, 105, 137, 171, 213, 218, 240, 320, 459, 503, 513, 525, 616, 761, 792, 833, 1048, 1127, 1257, 1369, 1395, 1536, 1551, 2025, 2457, 2600, 2610, 3267, 3312, 3600, 3681, 4032, 4100, 4125, 4128, 4201, 4901, 4976, 5001, 5225, 5880, 5975, 6167

Offset: 1

M. F. Hasler, Apr 06 2015

nonn

The sequence has the infinite subsequence (4^n*(2^n+16), n=0,1,2,...), with corresponding (A,B,C) = (2^(n+2),2^(n+1),2^n).

See A256652 for terms whose square has more than one representation of the given form. See A256613 for the subsequence of terms such that A^2 + B^3 + C^4 is a square, cf. A180241. See A256091 for the analog for sums of 3rd, 4th and 5th power.

			(A, B, C) = (1, 4, 2) = 1^4 + 4^5 + 2^6 = 1 + 1024 + 64 = 1089 = 33^2, so 33 is a term.
(A, B, C) = (1, 4, 8) = 1^4 + 4^5 + 8^6 = 1 + 1024 + 262144 = 263169 = 513^2, so 513 is a term.

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

is_A255830(D)=my(B,C=0,D2C6); while(1A255830(D)&&print1(D",")) \\ Converted to integer arithmetic by M. F. Hasler, May 01 2015

Inserted a(4)=33, a(18)=513 and removed doublet 1257 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

A180243 a(n) = number of ways in which the square of n can be expressed as the sum of a square, a cube and a fourth power.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 1, 1, 3, 0, 0, 2, 0, 0, 0, 0, 2, 1, 2, 0, 3, 1, 2, 1, 2, 0, 0, 3, 0, 0, 1, 1, 4, 0, 2, 0, 2, 2, 2, 0, 1, 2, 0, 1, 4, 2, 3, 0, 0, 0, 4, 1, 4, 1, 5, 0, 2, 0, 2, 2, 3, 0, 1, 0, 5, 3, 2, 3, 3, 3, 2, 2, 5, 1, 5, 0, 4, 1, 1, 2, 5, 3, 1, 3, 1, 0, 4, 2, 3, 1, 3, 0, 3, 0, 5, 1, 3, 2, 2, 2, 1, 0, 4, 0, 12

Offset: 1

Carmine Suriano, Aug 19 2010

nonn

The three terms of the sum and/or their bases can be the same.

			a(9)=3 since 9^2 = 81 = 1^2+4^3+2^4 = 4^2+4^3+1^4 = 8^2+1^3+2^4.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A180241, A180242.

A180244 Numbers whose square can be expressed as the sum of a square, a cube and a fourth power of three different numbers.

Original entry on oeis.org

5, 8, 9, 12, 17, 19, 21, 22, 23, 25, 28, 31, 32, 33, 35, 37, 38, 39, 42, 44, 45, 46, 47, 51, 52, 53, 54, 55, 57, 59, 60, 61, 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, 103, 105, 107

Offset: 1

Carmine Suriano, Aug 19 2010

nonn

Subset of A180241, containing terms arising from different bases only.

			a(5)=17 since 17^2 = 289 = 5^2+2^3+4^4 = 12^2+4^3+3^4; (5,2,4) and (12,4,3) are all different. [Corrected by _Bruno Berselli_, Aug 25 2010]

Cf. A180241, A180243.

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

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

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A345645 Numbers whose square can be represented in exactly one way as the sum of a square and a biquadrate (fourth power).

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A255830 Numbers D such that D^2 = A^4 + B^5 + C^6 for some positive integers A, B, C.

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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.

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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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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).