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.

Showing 1-6 of 6 results.

A350270 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

1, 1729, 5104, 4445, 4509, 4662, 5454, 6210, 9045, 11124, 14967, 17964, 22051, 26209, 32697, 39564, 46908, 56070, 66222, 78912, 92961, 105841, 123732, 143200, 162801, 188154, 212220, 241614, 271405, 307448, 344016, 383607, 428624, 475273, 529830, 586664, 645120
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 22 2021

Keywords

Examples

			For n = 2: 1729 = 1^3 + 12^3 = 9^3 + 10^3.
For n = 3: 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Links

Crossrefs

Cf. A000578, A025401, A025419, A025421, A275154, A350241.

Extensions

a(16)-a(27) from Michael S. Branicky, Dec 22 2021
More terms from Jinyuan Wang, Dec 30 2021

A360214 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero octahedral numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 231, 575, 721, 1618, 1750, 1877, 2240, 2736, 2995, 3105, 3080, 3500, 3311, 3920, 4151, 4280, 4495, 4719, 4621, 4675, 5041, 5164, 5291, 5060, 5591, 5480, 5566, 5635, 5755, 5985, 6216, 6080, 6279, 6320, 6510, 6655, 6636, 6870, 7145, 7195, 6999, 6971, 7296, 7211
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 30 2023

Keywords

Links

Crossrefs

Cf. A005900, A275154, A350205, A360215, A360216.

A360215 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero icosahedral numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 1383, 4157, 6548, 8633, 9884, 12503, 12920, 15357, 15812, 18146, 18126, 19755, 20895, 22106, 23229, 23246, 23685, 22118, 25142, 25884, 27894, 29448, 28149, 29703, 30285, 31914, 30966, 34007, 34380, 34390, 35082, 35894, 34389, 36891, 37035, 37425, 35907, 35895, 38856
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 30 2023

Keywords

Links

Crossrefs

Cf. A006564, A275154, A350205, A360214, A360216.

A360216 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero dodecahedral numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 2025, 2925, 9010, 15521, 18465, 19140, 24899, 32760, 33576, 36245, 39746, 39290, 39270, 46540, 50215, 49055, 53680, 50435, 56585, 58990, 57460, 58380, 61950, 63329, 64600, 63700, 64550, 67305, 68530, 71690
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 30 2023

Keywords

Links

Crossrefs

Cf. A006566, A275154, A350205, A360214, A360215.

A380448 Least primes which can be represented as the sum of distinct positive cubes in exactly n ways, or 0 if no such prime exists.

Original entry on oeis.org

73, 757, 953, 2521, 1801, 3257, 2953, 4013, 4139, 4789, 5347, 4481, 5669, 4663, 6427, 6659, 5867, 6301, 6841, 7867, 7687, 7741, 10169, 7057, 7723, 7561, 9631, 8443, 8191, 8387, 9883, 10079, 10313, 10891, 10729, 10009, 9109, 10711, 9829, 11161, 10457, 12547, 11699, 10513, 10333, 11159, 13007
Offset: 1

Views

Author

Zhining Yang, Jun 21 2025

Keywords

Comments

From David A. Corneth, Jun 22 2025: (Start)
a(234) = 0. We have if a(234) > 0 then a(234) > 10^6.
Proof:
Let Q(k, u) be the number of ways to write k as a sum of distinct cubes c where c <= u^3. Then for all m we have Q(k, m+1) >= Q(k, m).
for 39031 <= k <= 633371 we have Q(k, 40) >= 235.
Those are 633371 - 39031 + 1 = 594341 > 41^3 consecutive values for k.
Therefore Q(k, 41) >= 235 for 39031 <= k <= 633371 + 26^3.
We have n^3 + (n+1)^3 > (n+2)^3 for n >= 6. So via induction we have Q(k, u) > 234 for some u for any k >= 39031 which completes the proof.
Similarly Q(k, 40) > 10000 for 100000 <= k <= 500000 which is more than enough to confirm zero's found in the first 10000 terms in a search up to 10^6. (End)

Examples

			a(4) = 2521 because 2521 = 4^3 + 9^3 + 12^3 = 1^3 + 4^3 + 5^3 + 10^3 + 11^3= 1^3 +4^3+ 6^3 + 8^3 + 12^3 =4^3 + 6^3 + 8^3 + 9^3 + 10^3 and 2521 is the least prime that can be written as the sum of distinct positive cubes in 4 different ways.

Links

Crossrefs

Cf. A003997, A275154, A385094.

Programs

  • Mathematica
    s = CoefficientList[Series[Product[(1 + x^(r^3)), {r, 20}], {x, 0, 8000}], x];
Table[SelectFirst[Flatten@Position[s, k] - 1, PrimeQ], {k, 20}]

A333821 Numbers k that can be represented in the form k = p^3 - q^3 - r^3, where p, q, r are positive integers satisfying p = q + r.

Original entry on oeis.org

6, 18, 36, 48, 60, 90, 126, 144, 162, 168, 210, 216, 252, 270, 288, 330, 360, 378, 384, 396, 468, 480, 486, 540, 546, 594, 630, 720, 750, 792, 816, 858, 918, 924, 972, 990, 1008, 1026, 1140, 1152, 1170, 1260, 1296, 1344, 1386, 1404, 1518, 1530, 1560, 1620, 1638, 1656, 1680, 1728, 1800
Offset: 1

Views

Author

Antonio Roldán, Apr 06 2020

Keywords

Comments

An alternative representation of k is k = 3*q*r*(q+r), with q, r positive integers, then k is a multiple of 6.

Examples

			60 is in the sequence because 60 = 5^3 - 4^3 - 1^3, with 5 = 4 + 1.

Crossrefs

Cf. A000578, A003072, A024975, A078129, A255018, A275154.

Programs

  • PARI
    ok(n) = {my(i=1, a=0, m=0, j); if(n%6==0, while(a<=n&&m==0, j=1; while(j

Formula

a(n) = 6 * A121741(n).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.