A344799 -id:A344799 - cp's OEIS frontend

A025371 Numbers that are the sum of 4 nonzero squares in 6 or more ways.

90, 124, 130, 133, 135, 138, 147, 148, 150, 154, 156, 157, 159, 162, 163, 165, 166, 170, 171, 172, 174, 175, 177, 178, 180, 182, 183, 186, 187, 188, 189, 190, 193, 195, 196, 198, 199, 201, 202, 203, 205, 207, 210, 213, 214, 215, 217, 218, 219, 220, 222, 223, 225, 226

Offset: 1

David W. Wilson

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A025334, A025362, A025370, A025372, A344799, A345148.

limit = 226
from functools import lru_cache
sq = [k**2 for k in range(1, int(limit**.5)+2) if k**2 + 3 <= limit]
sqs = set(sq)
@lru_cache(maxsize=None)
def findsums(n, m):
  if m == 1: return {(n, )} if n in sqs else set()
  return set(tuple(sorted(t+(s,))) for s in sqs for t in findsums(n-s, m-1))
print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 6]) # Michael S. Branicky, Apr 20 2021

{n: A025428(n) >= 6}. Union of A025372 and A025362. - R. J. Mathar, Jun 15 2018

A345174 Numbers that are the sum of five third powers in six or more ways.

Original entry on oeis.org

2430, 2979, 3214, 3249, 3312, 3492, 3520, 3737, 3753, 3788, 3816, 3842, 3942, 3968, 4121, 4185, 4213, 4267, 4355, 4392, 4411, 4418, 4446, 4453, 4456, 4465, 4472, 4482, 4509, 4544, 4563, 4600, 4626, 4663, 4670, 4723, 4753, 4896, 4905, 4915, 4924, 4938, 4941

Offset: 1

David Consiglio, Jr., Jun 10 2021

nonn

			2430 is a term because 2430 = 1^3 + 2^3 + 2^3 + 5^3 + 12^3  = 1^3 + 3^3 + 4^3 + 7^3 + 11^3  = 2^3 + 2^3 + 6^3 + 6^3 + 11^3  = 2^3 + 3^3 + 3^3 + 9^3 + 10^3  = 3^3 + 5^3 + 8^3 + 8^3 + 8^3  = 3^3 + 4^3 + 7^3 + 8^3 + 9^3.

David Consiglio, Jr., Table of n, a(n) for n = 1..10000

Cf. A343989, A344799, A344940, A345148, A345175, A345180, A345515.

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, 5):
    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])

A344798 Numbers that are the sum of five squares in five or more ways.

Original entry on oeis.org

53, 56, 59, 61, 64, 67, 68, 74, 77, 79, 80, 83, 85, 86, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103, 104, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 130, 131, 133, 134, 135, 136, 137

Offset: 1

Sean A. Irvine, May 28 2021

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A025370, A294739, A343989, A344797, A344799, A344809.

A344800 Numbers that are the sum of five squares in seven or more ways.

Original entry on oeis.org

77, 83, 85, 88, 91, 94, 99, 101, 104, 106, 107, 109, 112, 115, 116, 118, 119, 120, 122, 123, 124, 125, 126, 127, 128, 130, 131, 133, 134, 136, 137, 138, 139, 140, 141, 142, 143, 144, 146, 147, 148, 149, 150, 151, 152, 154, 155, 156, 157, 158, 159, 160, 161

Offset: 1

Sean A. Irvine, May 28 2021

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A025372, A294741, A344799, A344801, A344811, A345180.

A344810 Numbers that are the sum of six squares in six or more ways.

Original entry on oeis.org

54, 57, 60, 62, 65, 68, 69, 71, 72, 75, 76, 77, 78, 80, 81, 83, 84, 86, 87, 88, 89, 90, 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, 121, 122, 123, 124, 125, 126, 127

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			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.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A344799, A344809, A344811, A345483, A345515.

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 >= 6])
    for x in range(len(rets)):
        print(rets[x])

Conjectures from Chai Wah Wu, Apr 25 2024: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 24.
G.f.: x*(-x^23 + x^22 - x^18 + x^17 - x^16 + x^15 - x^14 + x^13 - 2*x^10 + 2*x^9 - x^8 + x^7 - 2*x^6 + x^4 - x^3 - 51*x + 54)/(x - 1)^2. (End)

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A025371 Numbers that are the sum of 4 nonzero squares in 6 or more ways.

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A345174 Numbers that are the sum of five third powers in six or more ways.

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A344798 Numbers that are the sum of five squares in five or more ways.

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A344800 Numbers that are the sum of five squares in seven or more ways.

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A344810 Numbers that are the sum of six squares in six or more ways.

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