A345148 -id:A345148 - 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

A343987 Numbers that are the sum of four positive cubes in five or more ways.

Original entry on oeis.org

5105, 5131, 5616, 5859, 6435, 6883, 7777, 9315, 9737, 9793, 10017, 10250, 10458, 10936, 10962, 11000, 11060, 11088, 11592, 11664, 11781, 12168, 12229, 12285, 12320, 12385, 12392, 12411, 12707, 13104, 13384, 13734, 13832, 13904, 13923, 14112, 14183, 14239, 14581, 14833, 14896, 14904, 15176, 15561, 15596

Offset: 1

David Consiglio, Jr., May 06 2021

nonn

			5616 = 1^3 + 8^3 + 12^3 + 15^3
     = 2^3 + 8^3 + 10^3 + 16^3
     = 4^3 + 4^3 + 14^3 + 14^3
     = 4^3 + 5^3 + 11^3 + 16^3
     = 8^3 + 9^3 + 10^3 + 15^3
so 5616 is a term.

David Consiglio, Jr., Table of n, a(n) for n = 1..20000

Cf. A025370, A343967, A343971, A343986, A343989, A344356, A345148.

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, 50)]
for pos in cwr(power_terms, 4):
    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], end=", ")

A345150 Numbers that are the sum of four third powers in seven or more ways.

Original entry on oeis.org

13104, 18928, 19376, 20755, 21203, 21896, 22743, 24544, 24570, 24787, 25172, 25928, 27720, 27755, 27846, 28917, 29582, 30429, 31031, 31248, 31339, 31402, 31528, 32858, 33579, 34056, 34624, 34713, 34776, 35289, 35317, 35441, 35497, 35712, 36162, 36190, 36225

Offset: 1

David Consiglio, Jr., Jun 09 2021

nonn

			13104 is a term because 13104 = 1^3 + 10^3 + 16^3 + 18^3  = 1^3 + 11^3 + 14^3 + 19^3  = 2^3 + 9^3 + 15^3 + 19^3  = 4^3 + 6^3 + 14^3 + 20^3  = 4^3 + 9^3 + 10^3 + 21^3  = 5^3 + 7^3 + 11^3 + 21^3  = 8^3 + 9^3 + 14^3 + 19^3.

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

Cf. A025372, A344922, A345086, A345148, A345151, A345152, A345180.

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, 4):
    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])

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])

A344904 Numbers that are the sum of four fourth powers in six or more ways.

Original entry on oeis.org

3847554, 5624739, 6044418, 6576339, 6593538, 6899603, 9851058, 10456338, 11645394, 12378018, 13155858, 13638738, 16020018, 16408434, 16990803, 19081089, 20622338, 20649603, 20755218, 20795763, 22673634, 23056803, 24174003, 24368769, 25265553, 25850178

Offset: 1

David Consiglio, Jr., Jun 02 2021

nonn

			3847554 is a term because 3847554 = 2^4 + 13^4 + 29^4 + 42^4  = 2^4 + 21^4 + 22^4 + 43^4  = 6^4 + 11^4 + 17^4 + 44^4  = 6^4 + 31^4 + 32^4 + 37^4  = 9^4 + 29^4 + 32^4 + 38^4  = 13^4 + 26^4 + 32^4 + 39^4.

David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Cf. A344356, A344647, A344921, A344922, A344940, A345148.

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, 4):
    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])

A345083 Numbers that are the sum of three third powers in six or more ways.

Original entry on oeis.org

1296378, 1371735, 1409400, 1614185, 1824040, 1885248, 2016496, 2101464, 2302028, 2305395, 2542968, 2562624, 2851848, 2889216, 2974392, 2988441, 3185792, 3380833, 3681280, 3689496, 3706984, 3775680, 3906657, 4109832, 4123008, 4142683, 4422592, 4525632, 4783680

Offset: 1

David Consiglio, Jr., Jun 07 2021

nonn

			1296378 is a term because 1296378 = 3^3 + 75^3 + 94^3  = 8^3 + 32^3 + 107^3  = 20^3 + 76^3 + 93^3  = 30^3 + 58^3 + 101^3  = 32^3 + 80^3 + 89^3  = 59^3 + 74^3 + 86^3.

David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Cf. A025334, A343967, A344647, A345084, A345086, A345148.

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, 3):
    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])

A345149 Numbers that are the sum of four third powers in exactly six ways.

Original entry on oeis.org

6883, 12411, 13923, 14112, 14581, 14896, 14904, 15561, 15876, 16317, 16640, 17208, 17479, 17992, 18739, 18865, 19035, 19080, 19665, 19712, 19763, 19880, 20007, 20384, 20979, 21231, 21420, 21707, 22409, 22617, 23149, 23940, 24355, 25515, 25984, 26208, 26334

Offset: 1

David Consiglio, Jr., Jun 09 2021

nonn

Differs from A345148 at term 3 because 13104 = 1^3 + 10^3 + 16^3 + 18^3 = 1^3 + 11^3 + 14^3 + 19^3 = 2^3 + 9^3 + 15^3 + 19^3 = 4^3 + 6^3 + 14^3 + 20^3 = 4^3 + 9^3 + 10^3 + 21^3 = 5^3 + 7^3 + 11^3 + 21^3 = 8^3 + 9^3 + 14^3 + 19^3.

			6883 is a term because 6883 = 2^3 + 2^3 + 2^3 + 18^3  = 2^3 + 4^3 + 14^3 + 14^3  = 3^3 + 7^3 + 7^3 + 17^3  = 3^3 + 10^3 + 13^3 + 13^3  = 4^3 + 10^3 + 10^3 + 15^3  = 7^3 + 8^3 + 8^3 + 16^3.

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

Cf. A025362, A343986, A344921, A345084, A345148, A345151, A345175.

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, 4):
    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])

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

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A343987 Numbers that are the sum of four positive cubes in five or more ways.

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A345150 Numbers that are the sum of four third powers in seven 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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A344904 Numbers that are the sum of four fourth powers in six or more ways.

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A345083 Numbers that are the sum of three third powers in six or more ways.

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A345149 Numbers that are the sum of four third powers in exactly six ways.

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