cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A003335 Numbers that are the sum of 12 positive cubes.

Original entry on oeis.org

12, 19, 26, 33, 38, 40, 45, 47, 52, 54, 59, 61, 64, 66, 68, 71, 73, 75, 78, 80, 82, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 104, 106, 108, 110, 111, 113, 115, 116, 117, 118, 120, 122, 123, 124, 125, 127, 129, 130, 131, 132, 134, 136, 137, 138, 139, 141, 142
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
1120 is in the sequence as 1120 = 2^3 + 3^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 5^3 +  8^3.
2339 is in the sequence as 2339 = 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 9^3 +  9^3.
3594 is in the sequence as 3594 = 4^3 + 5^3 + 6^3 + 6^3 + 6^3 + 6^3 + 7^3 + 7^3 + 7^3 + 8^3 + 10^3. (End)

Links

Crossrefs

A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
Cf. A000578 (cubes).

Programs

  • PARI
    (A003335_upto(N, k=12, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ Use 2nd & 3rd optional arg to get other sequences of this family. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

A003333 Numbers that are the sum of 10 positive cubes.

Original entry on oeis.org

10, 17, 24, 31, 36, 38, 43, 45, 50, 52, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 108, 109, 111, 113, 114, 115, 116, 118, 120, 121, 122, 123, 125, 127, 128, 129, 130, 132, 134, 135, 136, 137, 139, 140, 141, 142
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

374 is the largest of only 99 positive integers not in this sequence. - M. F. Hasler, Aug 13 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
1646 is in the sequence as 1646 = 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 7^3 + 7^3 +  8^3.
2790 is in the sequence as 2790 = 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 6^3 + 6^3 + 7^3 + 8^3 + 10^3.
3450 is in the sequence as 3450 = 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 7^3 + 8^3 + 8^3 + 9^3 +  9^3. (End)

Links

Crossrefs

Other sequences of numbers that are the sum of x nonzero y-th powers:
A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

Programs

  • PARI
    (A003333_upto(N)=select( {is_A003333(n,k=10,m=3,L=sqrtnint(abs(n-k+1),m))=if( n>k*L^m || nM. F. Hasler, Aug 02 2020
A3333=A003333_upto(320); A003333(n)=if(n>275, n+99, n>222, n+98, A3333[n]) \\ M. F. Hasler, Aug 13 2020

Formula

a(n) = n + 99 for all n > 275. - M. F. Hasler, Aug 13 2020

A003368 Numbers that are the sum of 12 positive 6th powers.

Original entry on oeis.org

12, 75, 138, 201, 264, 327, 390, 453, 516, 579, 642, 705, 740, 768, 803, 866, 929, 992, 1055, 1118, 1181, 1244, 1307, 1370, 1433, 1468, 1531, 1594, 1657, 1720, 1783, 1846, 1909, 1972, 2035, 2098, 2196, 2259, 2322, 2385, 2448, 2511, 2574, 2637, 2700, 2763, 2924, 2987
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
54710 is in the sequence as 54710 = 2^6 + 3^6 + 3^6 + 3^6 + 3^6 + 4^6 + 4^6 + 4^6 + 4^6 + 4^6 + 5^6 + 5^6.
94302 is in the sequence as 94302 = 1^6 + 1^6 + 1^6 + 1^6 + 1^6 + 2^6 + 2^6 + 2^6 + 2^6 + 3^6 + 6^6 + 6^6.
133585 is in the sequence as 133585 = 1^6 + 1^6 + 1^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 4^6 + 4^6 + 4^6 + 7^6. (End)

Links

Crossrefs

Cf. A001014 (sixth powers).
Cf. A003358 - A003367 (numbers that are the sum of 2, ..., 11 positive 6th powers); A003335, A003346, A003357, A003379, A003390, A004801, A004812, A004823 (numbers that are the sum of 12 positive 3rd, ..., 11th powers).

Programs

  • Mathematica
    Module[{upto=2200,r},r=Ceiling[Surd[upto,6]];Select[Union[Total/@ Tuples[ Range[r]^6,12]],#<=upto&]] (* Harvey P. Dale, Aug 25 2015 *)
  • PARI
    (A003368_upto(N, k=12, m=6)=[n|n<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, n)])(3000) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

A003379 Numbers that are the sum of 12 positive 7th powers.

Original entry on oeis.org

12, 139, 266, 393, 520, 647, 774, 901, 1028, 1155, 1282, 1409, 1536, 2198, 2325, 2452, 2579, 2706, 2833, 2960, 3087, 3214, 3341, 3468, 3595, 4384, 4511, 4638, 4765, 4892, 5019, 5146, 5273, 5400, 5527, 5654, 6570, 6697, 6824, 6951, 7078, 7205, 7332, 7459, 7586, 7713
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
367634 is in the sequence as 367634 = 3^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 4^7 + 5^7 + 5^7 + 5^7.
456545 is in the sequence as 456545 = 2^7 + 2^7 + 2^7 + 4^7 + 4^7 + 4^7 + 4^7 + 5^7 + 5^7 + 5^7 + 5^7 + 5^7.
755708 is in the sequence as 755708 = 1^7 + 2^7 + 2^7 + 3^7 + 3^7 + 3^7 + 4^7 + 4^7 + 5^7 + 5^7 + 6^7 + 6^7. (End)

Links

Crossrefs

Cf. A001015 (seventh powers).
Cf. A003369 - A003378 (numbers that are the sum of 2, ..., 11 positive 7th powers); A003335, A003346, A003357, A003368, A003390, A004801, A004812, A004823 (numbers that are the sum of 12 positive 3rd, ..., 11th powers).

Programs

  • PARI
    (A003379_upto(N, k=12, m=7)=[n|n<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, n)])(8000) \\ 2nd & 3rd optional arg allow to get other sequences of this group. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020

Extensions

Offset corrected by David A. Corneth, Aug 03 2020

A003390 Sum of 12 nonzero 8th powers.

Original entry on oeis.org

12, 267, 522, 777, 1032, 1287, 1542, 1797, 2052, 2307, 2562, 2817, 3072, 6572, 6827, 7082, 7337, 7592, 7847, 8102, 8357, 8612, 8867, 9122, 9377, 13132, 13387, 13642, 13897, 14152, 14407, 14662, 14917, 15172, 15427, 15682, 19692, 19947, 20202, 20457
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
1890948 is in the sequence as 1890948 = 2^8 + 2^8 + 2^8 + 4^8 + 4^8 + 4^8 + 4^8 + 4^8 + 5^8 + 5^8 + 5^8 + 5^8.
2338951 is in the sequence as 2338951 = 1^8 + 1^8 + 1^8 + 1^8 + 1^8 + 3^8 + 4^8 + 4^8 + 4^8 + 4^8 + 5^8 + 6^8.
3841896 is in the sequence as 3841896 = 1^8 + 1^8 + 1^8 + 2^8 + 3^8 + 3^8 + 3^8 + 3^8 + 4^8 + 5^8 + 6^8 + 6^8. (End)

Links

Crossrefs

A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2), A003072 (3, 3), A003325 (2, 3), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003386 (8, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11).

Programs

  • PARI
    A003390_upto(N=1e5, n=12, p=8)={my(P=[x^p|x<-[1..sqrtnint(N-n+1, p)]], S=P); while(n--, S=Set(concat([[x+y|y<-S, x+y<=N]|x<-P]))); S} \\ M. F. Hasler, Jul 03 2025

Extensions

Removed incorrect program, offset corrected by David A. Corneth, Aug 01 2020

A004802 Numbers that are the sum of 2 nonzero 10th powers.

Original entry on oeis.org

2, 1025, 2048, 59050, 60073, 118098, 1048577, 1049600, 1107625, 2097152, 9765626, 9766649, 9824674, 10814201, 19531250, 60466177, 60467200, 60525225, 61514752, 70231801, 120932352, 282475250, 282476273, 282534298, 283523825, 292240874
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
1103972715709403850 is in the sequence as 1103972715709403850 = 51^10 + 63^10.
2059617246125773226 is in the sequence as 2059617246125773226 = 61^10 + 65^10.
27850192968371852849 is in the sequence as 27850192968371852849 = 25^10 + 88^10. (End)

Links

Crossrefs

A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Extensions

Removed incorrect program. - David A. Corneth, Aug 01 2020

A003338 Numbers that are the sum of 4 nonzero 4th powers.

Original entry on oeis.org

4, 19, 34, 49, 64, 84, 99, 114, 129, 164, 179, 194, 244, 259, 274, 289, 304, 324, 339, 354, 369, 419, 434, 499, 514, 529, 544, 594, 609, 628, 643, 658, 673, 674, 708, 723, 738, 769, 784, 788, 803, 849, 868, 883, 898, 913, 963, 978, 1024, 1043, 1138, 1153, 1218
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
53667 is in the sequence as 53667 = 2^4 + 5^4 + 7^4 + 15^4.
81427 is in the sequence as 81427 = 5^4 + 5^4 + 11^4 + 16^4.
106307 is in the sequence as 106307 = 3^4 + 5^4 + 5^4 + 18^4. (End)

Links

Crossrefs

Cf. A047715, A309763 (more than 1 way), A344189 (exactly 2 ways), A176197 (distinct nonzero powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Programs

  • Maple
    # returns number of ways of writing n as a^4+b^4+c^4+d^4, 1<=a<=b<=c<=d.
A003338 := proc(n)
    local a,i,j,k,l,res ;
    a := 0 ;
    for i from 1 do
        if i^4 > n then
            break ;
        end if;
        for j from i do
            if i^4+j^4 > n then
                break ;
            end if;
            for k from j do
                if i^4+j^4+k^4> n then
                    break;
                end if;
                res := n-i^4-j^4-k^4 ;
                if issqr(res) then
                    res := sqrt(res) ;
                    if issqr(res) then
                        l := sqrt(res) ;
                        if l >= k then
                            a := a+1 ;
                        end if;
                    end if;
                end if;
            end do:
        end do:
    end do:
    a ;
end proc:
for n from 1 do
    if A003338(n) > 0 then
        print(n) ;
    end if;
end do: # R. J. Mathar, May 17 2023
  • Mathematica
    f[maxno_]:=Module[{nn=Floor[Power[maxno-3, 1/4]],seq}, seq=Union[Total/@(Tuples[Range[nn],{4}]^4)]; Select[seq,#<=maxno&]]
f[1000] (* Harvey P. Dale, Feb 27 2011 *)
  • Python
    limit = 1218
from functools import lru_cache
qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit]
qds = set(qd)
@lru_cache(maxsize=None)
def findsums(n, m):
  if m == 1: return {(n, )} if n in qds else set()
  return set(tuple(sorted(t+(q,))) for q in qds for t in findsums(n-q, m-1))
print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1]) # Michael S. Branicky, Apr 19 2021

A003330 Numbers that are the sum of 7 positive cubes.

Original entry on oeis.org

7, 14, 21, 28, 33, 35, 40, 42, 47, 49, 54, 56, 59, 61, 66, 68, 70, 73, 75, 77, 80, 84, 85, 87, 91, 92, 94, 96, 98, 99, 103, 105, 106, 110, 111, 112, 113, 117, 118, 122, 124, 125, 129, 131, 132, 133, 136, 137, 138, 140, 143, 144, 145, 147, 148, 150, 151, 152, 154
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020
2408 is the largest among only 208 positive integers not in this sequence: cf. formula. - M. F. Hasler, Aug 23 2020

Examples

			From _M. F. Hasler_, Aug 23 2020: (Start)
The first few terms are multiples of 7 because of the coincidence that 2^3 - 1^3 = 7, equal to the number of cubes we consider here:
7 = 1^3 * 7 is the smallest sum of seven positive cubes.
14 = 1^3 * 6 + 2^3 = 6 + 8 is the next larger sum of seven positive cubes.
21 = 1^3 * 5 + 2^3 * 2 = 5 + 16 is the next larger sum of seven positive cubes.
28 = 1^3 * 4 + 2^3 * 3 = 4 + 24 is the next larger sum of seven positive cubes.
There are three more terms of this form, but the next larger sum of seven positive cubes is a(5) = 3^3 + 6 * 1^3 = 33. (End)
From _David A. Corneth_, Aug 01 2020: (Start)
2070 is in the sequence as 2070 = 4^3 + 4^3 + 4^3 + 5^3 + 8^3 + 8^3 +  9^3.
2383 is in the sequence as 2383 = 3^3 + 5^3 + 5^3 + 6^3 + 6^3 + 7^3 + 11^3.
3592 is in the sequence as 3592 = 4^3 + 5^3 + 6^3 + 9^3 + 9^3 + 9^3 + 10^3. (End)

Links

Crossrefs

Cf. A000578, A004829, A003331, A003341.
Other sequences of numbers that are the sum of x nonzero y-th powers:
A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

Programs

  • PARI
    (A003330_upto(N, k=7, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(160) \\ M. F. Hasler, Aug 02 2020

Formula

a(n) = n + 208 for all n > 2200. - M. F. Hasler, Aug 23 2020

Extensions

More terms from Arlin Anderson (starship1(AT)gmail.com)

A003331 Numbers that are the sum of 8 positive cubes.

Original entry on oeis.org

8, 15, 22, 29, 34, 36, 41, 43, 48, 50, 55, 57, 60, 62, 64, 67, 69, 71, 74, 76, 78, 81, 83, 85, 86, 88, 92, 93, 95, 97, 99, 100, 102, 104, 106, 107, 111, 112, 113, 114, 118, 119, 120, 121, 123, 125, 126, 130, 132, 133, 134, 137, 138, 139, 140, 141, 144, 145, 146, 148, 149
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

620 is the largest among only 142 positive integers not in this sequence. This can be proved by induction. - M. F. Hasler, Aug 13 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
1796 is in the sequence as 1796 = 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 7^3 + 7^3 + 9^3.
2246 is in the sequence as 2246 = 2^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3 + 7^3 + 11^3.
3164 is in the sequence as 3164 = 5^3 + 5^3 + 6^3 + 6^3 + 8^3 + 8^3 + 9^3 + 9^3.(End)

Links

Crossrefs

Other sequences of numbers that are the sum of x nonzero y-th powers:
A000404 (x=2, y=2), A000408 (3, 2), A000414 (4, 2), A047700 (5, 2),
A003325 (2, 3), A003072 (3, 3), A003327 .. A003335 (4 .. 12, 3),
A003336 .. A003346 (2 .. 12, 4), A003347 .. A003357 (2 .. 12, 5),
A003358 .. A003368 (2 .. 12, 6), A003369 .. A003379 (2 .. 12, 7),
A003380 .. A003390 (2 .. 12, 8), A003391 .. A004801 (2 .. 12, 9),
A004802 .. A004812 (2 .. 12, 10), A004813 .. A004823 (2 .. 12, 11).

Programs

  • Mathematica
    Module[{upto=200,c},c=Floor[Surd[upto,3]];Select[Union[Total/@ Tuples[ Range[ c]^3,8]],#<=upto&]] (* Harvey P. Dale, Jan 11 2016 *)
  • PARI
    (A003331_upto(N, k=8, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ M. F. Hasler, Aug 02 2020
A003331(n)=if(n>478, n+142, n>329, n+141, A003331_upto(470)[n]) \\ M. F. Hasler, Aug 13 2020
  • Python
    from itertools import combinations_with_replacement as mc
def aupto(lim):
    cbs = (i**3 for i in range(1, int((lim-7)**(1/3))+2))
    return sorted(set(k for k in (sum(c) for c in mc(cbs, 8)) if k <= lim))
print(aupto(150)) # Michael S. Branicky, Aug 15 2021

Formula

a(n) = 142 + n for all n > 478. - M. F. Hasler, Aug 13 2020

A003356 Numbers that are the sum of 11 positive 5th powers.

Original entry on oeis.org

11, 42, 73, 104, 135, 166, 197, 228, 253, 259, 284, 290, 315, 321, 346, 352, 377, 408, 439, 470, 495, 501, 526, 532, 557, 563, 588, 619, 650, 681, 712, 737, 743, 768, 774, 799, 830, 861, 892, 923, 954, 979, 985, 1010, 1034, 1041, 1065, 1072, 1096, 1103, 1127, 1134
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
16989 is in the sequence as 16989 = 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 4^5 + 5^5 + 5^5 + 5^5 + 5^5 + 5^5.
22564 is in the sequence as 22564 = 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 4^5 + 4^5 + 5^5 + 7^5.
30191 is in the sequence as 30191 = 1^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 4^5 + 5^5 + 6^5 + 7^5. (End)

Links

Crossrefs

Cf. A000584 (fifth powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Extensions

Incorrect program removed by David A. Corneth, Aug 01 2020
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