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.

A345515 Numbers that are the sum of six cubes in six or more ways.

Original entry on oeis.org

1377, 1488, 1586, 1595, 1647, 1673, 1677, 1710, 1738, 1764, 1766, 1773, 1799, 1829, 1836, 1837, 1862, 1881, 1890, 1911, 1953, 1955, 1981, 1988, 2007, 2011, 2014, 2018, 2025, 2044, 2051, 2070, 2079, 2097, 2105, 2107, 2108, 2142, 2153, 2160, 2168, 2170, 2177
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			1488 is a term because 1488 = 1^3 + 1^3 + 1^3 + 3^3 + 8^3 + 8^3 = 1^3 + 1^3 + 3^3 + 3^3 + 3^3 + 10^3 = 1^3 + 2^3 + 3^3 + 6^3 + 6^3 + 8^3 = 2^3 + 2^3 + 2^3 + 2^3 + 4^3 + 10^3 = 3^3 + 3^3 + 3^3 + 3^3 + 6^3 + 9^3 = 3^3 + 5^3 + 5^3 + 6^3 + 6^3 + 6^3.

Links

Crossrefs

Cf. A344810, A345174, A345514, A345516, A345524, A345563, A345768.

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, 6):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 6])
    for x in range(len(rets)):
        print(rets[x])

A344799 Numbers that are the sum of five squares in six or more ways.

Original entry on oeis.org

77, 80, 83, 85, 86, 88, 91, 92, 94, 98, 99, 100, 101, 103, 104, 106, 107, 109, 110, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 130, 131, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149
Offset: 1

Views

Author

Sean A. Irvine, May 28 2021

Keywords

Links

Crossrefs

Cf. A025371, A294740, A344798, A344800, A344810, A345174.

A344809 Numbers that are the sum of six squares in five or more ways.

Original entry on oeis.org

54, 57, 60, 62, 63, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			57 = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 7^2
   = 1^2 + 1^2 + 1^2 + 2^2 + 5^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 6^2
   = 1^2 + 2^2 + 2^2 + 4^2 + 4^2 + 4^2
   = 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 5^2
   = 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 5^2
so 57 is a term.

Links

Crossrefs

Cf. A344798, A344808, A344810, A345482, A345514.

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, 6):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 5])
    for x in range(len(rets)):
        print(rets[x])

A344811 Numbers that are the sum of six squares in seven or more ways.

Original entry on oeis.org

60, 65, 68, 69, 77, 78, 81, 84, 86, 87, 89, 90, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			65 = 1^2 + 1^2 + 1^2 + 1^2 + 5^2 + 6^2
   = 1^2 + 1^2 + 1^2 + 2^2 + 3^2 + 7^2
   = 1^2 + 1^2 + 2^2 + 3^2 + 5^2 + 5^2
   = 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 6^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 4^2 + 6^2
   = 2^2 + 2^2 + 3^2 + 4^2 + 4^2 + 4^2
   = 2^2 + 3^2 + 3^2 + 3^2 + 3^2 + 5^2
so 65 is a term.

Links

Crossrefs

Cf. A344800, A344810, A344812, A345484, A345516.

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, 6):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 7])
    for x in range(len(rets)):
        print(rets[x])

A345483 Numbers that are the sum of seven squares in six or more ways.

Original entry on oeis.org

55, 58, 61, 63, 64, 66, 67, 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, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			58 is a term because 58 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 7^2 = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 5^2 + 5^2 = 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 6^2 = 1^2 + 1^2 + 2^2 + 2^2 + 4^2 + 4^2 + 4^2 = 1^2 + 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 5^2 = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 5^2 = 2^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2.

Links

Crossrefs

Cf. A344810, A345482, A345484, A345493, A345524.

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, 7):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 6])
    for x in range(len(rets)):
        print(rets[x])

Formula

Conjectures from Chai Wah Wu, Apr 25 2024: (Start)
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 - 52*x + 55)/(x - 1)^2. (End)
Showing 1-5 of 5 results.

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

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