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 1-5 of 5 results.

A345541 Numbers that are the sum of nine cubes in two or more ways.

Original entry on oeis.org

72, 133, 140, 147, 159, 161, 166, 168, 175, 182, 185, 187, 189, 194, 196, 198, 201, 203, 205, 208, 213, 217, 220, 222, 224, 227, 231, 238, 239, 243, 245, 246, 250, 252, 257, 259, 261, 264, 265, 266, 271, 273, 276, 278, 280, 283, 285, 287, 289, 290, 292, 294
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			133 is a term because 133 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 4^3 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 3^3.

Links

Crossrefs

Cf. A003332, A345499, A345532, A345542, A345550, A345586, A345794.

Programs

  • Python
    from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1, 1000)]
for pos in cwr(power_terms, 9):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 2])
    for x in range(len(rets)):
        print(rets[x])

A345551 Numbers that are the sum of ten cubes in three or more ways.

Original entry on oeis.org

197, 225, 232, 239, 246, 251, 253, 258, 260, 265, 267, 272, 277, 279, 281, 284, 286, 288, 291, 293, 295, 298, 300, 302, 303, 305, 307, 309, 310, 312, 314, 316, 317, 319, 321, 323, 324, 326, 328, 329, 330, 335, 336, 338, 340, 342, 343, 344, 345, 347, 349, 351
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			225 is a term because 225 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 5^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 3^3 + 3^3 + 3^3 = 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 = 1^3 + 2^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3.

Links

Crossrefs

Cf. A345510, A345542, A345550, A345552, A345596, A345805.

Programs

  • Python
    from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1, 1000)]
for pos in cwr(power_terms, 10):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 3])
    for x in range(len(rets)):
        print(rets[x])

A345595 Numbers that are the sum of ten fourth powers in two or more ways.

Original entry on oeis.org

265, 280, 295, 310, 325, 340, 345, 355, 360, 375, 390, 405, 420, 425, 440, 455, 470, 485, 505, 520, 535, 550, 565, 580, 585, 595, 600, 615, 630, 645, 660, 665, 680, 695, 710, 725, 745, 760, 775, 790, 805, 820, 835, 840, 855, 870, 885, 889, 900, 904, 919, 920
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			280 is a term because 280 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 4^4 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4.

Links

Crossrefs

Cf. A003344, A345550, A345586, A345596, A345634, A345854.

Programs

  • Python
    from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**4 for x in range(1, 1000)]
for pos in cwr(power_terms, 10):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 2])
    for x in range(len(rets)):
        print(rets[x])

A345804 Numbers that are the sum of ten cubes in exactly two ways.

Original entry on oeis.org

73, 80, 99, 134, 136, 141, 148, 155, 160, 162, 167, 169, 174, 176, 183, 186, 188, 190, 192, 193, 195, 199, 202, 204, 206, 209, 211, 212, 213, 214, 216, 218, 221, 223, 228, 230, 235, 240, 244, 247, 249, 254, 262, 266, 269, 270, 273, 274, 290, 292, 297, 304
Offset: 1

Views

Author

David Consiglio, Jr., Jun 26 2021

Keywords

Comments

Differs from A345550 at term 22 because 197 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 4^3 + 5^3 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 + 4^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 5^3.
Likely finite.

Examples

			80 is a term because 80 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 3^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3.

Links

Crossrefs

Cf. A345550, A345794, A345803, A345805, A345854.

Programs

  • Python
    from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1, 1000)]
for pos in cwr(power_terms, 10):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v == 2])
    for x in range(len(rets)):
        print(rets[x])

A345509 Numbers that are the sum of ten squares in two or more ways.

Original entry on oeis.org

25, 28, 31, 33, 34, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			28 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 4^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2
so 28 is a term.

Links

Crossrefs

Cf. A025429, A345499, A345508, A345510, A345550.

Programs

  • Python
    from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**2 for x in range(1, 1000)]
for pos in cwr(power_terms, 10):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 2])
    for x in range(len(rets)):
        print(rets[x])
  • Python
    def A345509(n): return (25, 28, 31, 33, 34, 36, 37)[n-1] if n<8 else n+31 # Chai Wah Wu, May 09 2024

Formula

From Chai Wah Wu, May 09 2024: (Start)
All integers >= 39 are terms. Proof: since 20 can be written as the sum of 5 positive squares in 2 ways and any integer >= 34 can be written as a sum of 5 positive squares (see A025429), any integer >= 54 can be written as a sum of 10 positive squares in 2 or more ways. Integers from 39 to 53 are terms by inspection.
a(n) = 2*a(n-1) - a(n-2) for n > 9.
G.f.: x*(-x^8 + x^7 - x^6 + x^5 - x^4 - x^3 - 22*x + 25)/(x - 1)^2. (End)
Showing 1-5 of 5 results.

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