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.

A345522 Numbers that are the sum of seven cubes in four or more ways.

Original entry on oeis.org

470, 496, 503, 603, 627, 634, 653, 659, 685, 690, 692, 711, 712, 747, 751, 754, 761, 766, 768, 773, 775, 777, 780, 783, 787, 792, 794, 812, 813, 829, 831, 836, 838, 842, 843, 845, 857, 859, 864, 867, 871, 874, 875, 881, 883, 885, 890, 892, 894, 899, 900, 901
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			496 is a term because 496 = 1^3 + 1^3 + 1^3 + 3^3 + 4^3 + 4^3 + 5^3 = 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 5^3 + 5^3 = 1^3 + 2^3 + 2^3 + 2^3 + 3^3 + 3^3 + 6^3 = 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 4^3 + 4^3.

Links

Crossrefs

Cf. A345481, A345513, A345521, A345523, A345534, A345570, A345776.

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, 7):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 4])
    for x in range(len(rets)):
        print(rets[x])

A344808 Numbers that are the sum of six squares in four or more ways.

Original entry on oeis.org

36, 41, 44, 45, 53, 54, 56, 57, 60, 62, 63, 65, 66, 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, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			41 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 6^2
   = 1^2 + 1^2 + 1^2 + 2^2 + 3^2 + 5^2
   = 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 3^2
   = 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 4^2
so 41 is a term.

Links

Crossrefs

Cf. A344797, A344807, A344809, A345481, A345513.

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 >= 4])
    for x in range(len(rets)):
        print(rets[x])

A345480 Numbers that are the sum of seven squares in three or more ways.

Original entry on oeis.org

31, 34, 37, 39, 40, 42, 43, 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, 97, 98, 99, 100, 101
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			34 is a term because 34 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 5^2 = 1^2 + 1^2 + 1^2 + 2^2 + 3^2 + 3^2 + 3^2 = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2.

Links

Crossrefs

Cf. A344807, A345479, A345481, A345490, A345521.

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 >= 3])
    for x in range(len(rets)):
        print(rets[x])

A345482 Numbers that are the sum of seven squares in five or more ways.

Original entry on oeis.org

45, 54, 55, 57, 58, 60, 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
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			54 = 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 4^2 + 5^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 6^2
   = 1^2 + 1^2 + 3^2 + 3^2 + 3^2 + 3^2 + 4^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 3^2 + 4^2 + 4^2
   = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 5^2
so 54 is a term.

Links

Crossrefs

Cf. A344809, A345481, A345483, A345492, A345523.

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 >= 5])
    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 > 13.
G.f.: x*(-x^12 + x^11 - x^10 + x^9 - x^8 + x^7 - x^6 + x^5 - x^4 + x^3 - 8*x^2 - 36*x + 45)/(x - 1)^2. (End)

A345491 Numbers that are the sum of eight squares in four or more ways.

Original entry on oeis.org

32, 38, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 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, 97, 98, 99, 100, 101, 102, 103, 104, 105
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			38 is a term because 38 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 + 4^2 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 5^2 = 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 3^2 + 3^2 + 3^2 = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2.

Links

Crossrefs

Cf. A345481, A345490, A345492, A345501, A345534.

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, 8):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 4])
    for x in range(len(rets)):
        print(rets[x])
Showing 1-5 of 5 results.

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