A344355 -id:A344355 - cp's OEIS frontend

A344244 Numbers that are the sum of five fourth powers in exactly three ways.

4225, 6610, 6850, 9170, 9235, 9490, 11299, 12929, 14209, 14690, 14755, 14770, 15314, 16579, 16594, 16659, 16834, 17203, 17235, 17315, 17859, 17874, 17939, 18785, 18850, 18979, 19154, 19700, 19715, 20674, 21250, 21330, 21364, 21410, 21954, 23139, 23795, 24754, 25810, 26578, 28610, 28930, 29330, 29699

Offset: 1

David Consiglio, Jr., May 12 2021

nonn

Differs from A344243 at term 31 because 20995 = 1^4 + 1^4 + 1^4 + 4^4 + 12^4 = 2^4 + 3^4 + 3^4 + 3^4 + 12^4 = 2^4 + 6^4 + 9^4 + 9^4 + 9^4 = 4^4 + 6^4 + 7^4 + 7^4 + 11^4

			6850 is a member of this sequence because 6850 =  = 1^4 + 2^4 + 2^4 + 4^4 + 9^4 = 2^4 + 3^4 + 4^4 + 7^4 + 8^4 = 3^4 + 3^4 + 6^4 + 6^4 + 8^4

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

Cf. A342688, A343705, A344237, A344242, A344243, A344355, A345815.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**4 for x in range(1,50)]
for pos in cwr(power_terms,5):
    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])

A344359 Numbers that are the sum of five fourth powers in exactly five ways.

Original entry on oeis.org

59779, 67859, 93394, 108274, 112850, 136915, 142354, 151475, 161459, 168979, 181219, 183539, 183604, 185299, 187699, 189394, 193379, 195394, 199090, 199474, 200979, 201874, 202979, 203299, 205859, 211330, 212419, 213730, 217810, 217890, 221779, 223090, 223155, 223714, 226514, 227779, 231235

Offset: 1

David Consiglio, Jr., May 15 2021

nonn

Differs from A344358 at term 8 because 151300 = 3^4 + 3^4 + 3^4 + 12^4 + 19^4 = 3^4 + 11^4 + 11^4 + 14^4 + 17^4 = 3^4 + 13^4 + 13^4 + 13^4 + 16^4 = 6^4 + 9^4 + 9^4 + 9^4 + 19^4 = 7^4 + 11^4 + 11^4 + 11^4 + 18^4 = 8^4 + 9^4 + 13^4 + 13^4 + 17^4.

			93394 is a term of this sequence because 93394 = 1^4 + 4^4 + 8^4 + 14^4 + 15^4 = 1^4 + 6^4 + 12^4 + 12^4 + 15^4 = 1^4 + 9^4 + 10^4 + 14^4 + 14^4 = 5^4 + 6^4 + 11^4 + 14^4 + 14^4 = 5^4 + 7^4 + 8^4 + 12^4 + 16^4.

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

Cf. A343988, A344355, A344357, A344358, A344941, A345817, A346257.

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, 50)]
for pos in cwr(power_terms, 5):
    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])

A345816 Numbers that are the sum of six fourth powers in exactly four ways.

Original entry on oeis.org

6626, 6691, 6866, 9251, 9491, 10115, 10706, 10786, 11555, 12595, 14225, 14691, 14771, 15315, 15330, 15570, 16051, 16595, 16660, 16675, 16850, 17090, 17091, 17236, 17316, 17331, 17346, 17860, 17875, 17940, 17955, 18195, 18786, 18851, 19155, 19170, 19475, 19490

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345561 at term 16 because 15395 = 1^4 + 1^4 + 1^4 + 6^4 + 8^4 + 10^4 = 1^4 + 2^4 + 5^4 + 8^4 + 8^4 + 9^4 = 3^4 + 4^4 + 4^4 + 7^4 + 7^4 + 10^4 = 3^4 + 5^4 + 7^4 + 8^4 + 8^4 + 8^4 = 2^4 + 2^4 + 2^4 + 3^4 + 5^4 + 11^4.

			6691 is a term because 6691 = 1^4 + 1^4 + 1^4 + 6^4 + 6^4 + 8^4 = 1^4 + 2^4 + 2^4 + 2^4 + 3^4 + 9^4 = 2^4 + 2^4 + 3^4 + 3^4 + 7^4 + 8^4 = 3^4 + 4^4 + 4^4 + 6^4 + 7^4 + 7^4.

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

Cf. A344355, A345561, A345766, A345815, A345817, A345826, A346359.

Select[Range[20000],Count[PowersRepresentations[#,6,4],?(#[[1]]>0&)]==4&] (* _Harvey P. Dale, Mar 11 2023 *)

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, 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])

A344035 Numbers that are the sum of five positive cubes in exactly four ways.

Original entry on oeis.org

1252, 1376, 1461, 1522, 1548, 1585, 1590, 1646, 1702, 1709, 1737, 1739, 1772, 1798, 1802, 1810, 1864, 1889, 1954, 1987, 2006, 2033, 2081, 2096, 2152, 2160, 2225, 2241, 2251, 2276, 2313, 2322, 2339, 2341, 2367, 2374, 2377, 2416, 2423, 2456, 2458, 2465, 2467, 2512, 2521, 2528, 2530, 2537, 2540, 2549, 2556, 2582

Offset: 1

David Consiglio, Jr., May 07 2021

nonn

Differs from A344034 at term 13 because 1765 = 1^3 + 1^3 + 2^3 + 3^3 + 12^3 = 1^3 + 1^3 + 6^3 + 6^3 + 11^3 = 1^3 + 2^3 + 3^3 + 9^3 + 10^3 = 3^3 + 4^3 + 6^3 + 9^3 + 9^3 = 4^3 + 4^3 + 5^3 + 8^3 + 10^3

			1461 is a member of this sequence because 1461 = 1^3 + 1^3 + 1^3 + 9^3 + 9^3 = 1^3 + 1^3 + 4^3 + 4^3 + 11^3 = 3^3 + 3^3 + 4^3 + 7^3 + 10^3 = 6^3 + 6^3 + 7^3 + 7^3 + 7^3

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

Cf. A294738, A343705, A343972, A343988, A344034, A344355, A345766.

s5pcQ[n_]:=Length[Select[PowersRepresentations[n,5,3],FreeQ[#,0]&]]==4; Select[Range[ 3000],s5pcQ] (* Harvey P. Dale, Sep 15 2024 *)

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,5):
    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])

A344353 Numbers that are the sum of four fourth powers in exactly four ways.

Original entry on oeis.org

236674, 282018, 300834, 334818, 478338, 637794, 650034, 650658, 708483, 708834, 729938, 789378, 816578, 832274, 849954, 941859, 989043, 1042083, 1045539, 1099203, 1099458, 1102258, 1179378, 1243074, 1257954, 1283874, 1323234, 1334979, 1339074, 1342979, 1352898, 1357059, 1379043, 1518578

Offset: 1

David Consiglio, Jr., May 15 2021

nonn

Differs from A344352 at term 52 because 2147874 = 2^4 + 14^4 + 31^4 + 33^4 = 4^4 + 23^4 + 27^4 + 34^4 = 7^4 + 21^4 + 28^4 + 34^4 = 12^4 + 17^4 + 29^4 + 34^4 = 14^4 + 18^4 + 19^4 + 37^4.

			300834 is a term of this sequence because 300834 = 1^4 + 4^4 + 12^4 + 23^4 = 1^4 + 16^4 + 18^4 + 19^4 = 3^4 + 6^4 + 18^4 + 21^4 = 7^4 + 14^4 + 16^4 + 21^4.

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

Cf. A343972, A344242, A344278, A344352, A344355, A344357.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**4 for x in range(1,200)]
count = 1
for pos in cwr(power_terms,4):
    tot = sum(pos)
    keep[tot] += 1
    count += 1
rets = sorted([k for k,v in keep.items() if v == 4])
for x in range(len(rets)):
    print(rets[x])

A344354 Numbers that are the sum of five fourth powers in four or more ways.

Original entry on oeis.org

20995, 21235, 31250, 41474, 43235, 43250, 43315, 43490, 43859, 45139, 46290, 47570, 51939, 53234, 53299, 54994, 56274, 57379, 57410, 57779, 59329, 59779, 63970, 67010, 67859, 68035, 68290, 71795, 71954, 73730, 73954, 75714, 75794, 77890, 82099, 84499, 86275, 86450, 87730, 92500, 93394, 93474, 93859

Offset: 1

David Consiglio, Jr., May 15 2021

nonn

			31250 is a term of this sequence because 31250 = 2^4 + 2^4 + 4^4 + 7^4 + 13^4 = 2^4 + 3^4 + 6^4 + 6^4 + 13^4 = 4^4 + 6^4 + 7^4 + 9^4 + 12^4 = 5^4 + 5^4 + 10^4 + 10^4 + 10^4.

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

Cf. A344034, A344243, A344352, A344355, A344358, A344518, A345561.

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, 50)]
for pos in cwr(power_terms, 5):
    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])

A344519 Numbers that are the sum of five positive fifth powers in exactly four ways.

Original entry on oeis.org

287618651, 1386406515, 1763135232, 2494769760, 2619898293, 3096064443, 3291315732, 3749564512, 4045994624, 5142310350, 5183605813, 5658934676, 5880926107, 7205217018, 7401155424, 7691215599, 8429499101, 8926086432, 9051501568, 9203796832, 9254212901

Offset: 1

David Consiglio, Jr., May 21 2021

nonn

Differs from A344518 at term 20 because
9006349824 =  8^5 + 34^5 + 62^5 + 68^5 + 92^5
           =  8^5 + 41^5 + 47^5 + 79^5 + 89^5
           = 12^5 + 18^5 + 72^5 + 78^5 + 84^5
           = 21^5 + 34^5 + 43^5 + 74^5 + 92^5
           = 24^5 + 42^5 + 48^5 + 54^5 + 96^5.

			287618651 is a term because
287618651 =  8^5 + 21^5 + 27^5 + 27^5 + 48^5
          =  9^5 + 13^5 + 26^5 + 37^5 + 46^5
          = 11^5 + 12^5 + 23^5 + 41^5 + 44^5
          = 11^5 + 20^5 + 22^5 + 30^5 + 48^5.
[Corrected by _Patrick De Geest_, Dec 28 2024]

Sean A. Irvine, Table of n, a(n) for n = 1..4857

Cf. A003350, A004845, A342685, A342686, A342687, A342688, A344244, A344355, A344518, A345863, A345864, A346257, A346358, A346359.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**5 for x in range(1, 500)]
for pos in cwr(power_terms, 5):
    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])

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A344244 Numbers that are the sum of five fourth powers in exactly three ways.

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A344359 Numbers that are the sum of five fourth powers in exactly five ways.

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A345816 Numbers that are the sum of six fourth powers in exactly four ways.

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A344035 Numbers that are the sum of five positive cubes in exactly four ways.

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A344353 Numbers that are the sum of four fourth powers in exactly four ways.

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A344354 Numbers that are the sum of five fourth powers in four or more ways.

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A344519 Numbers that are the sum of five positive fifth powers in exactly four ways.

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