A344241 -id:A344241 - cp's OEIS frontend

A309763 Numbers that are the sum of 4 nonzero 4th powers in more than one way.

259, 2674, 2689, 2754, 2929, 3298, 3969, 4144, 4209, 5074, 6579, 6594, 6659, 6769, 6834, 7203, 7874, 8194, 8979, 9154, 9234, 10113, 10674, 11298, 12673, 12913, 13139, 14674, 14689, 14754, 16563, 16578, 16643, 16818, 17187, 17234, 17299, 17314, 17858, 18963, 19699

Offset: 1

Ilya Gutkovskiy, Aug 15 2019

nonn

			259 = 1^4 + 1^4 + 1^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4, so 259 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A003338, A018786, A025367, A025406, A309762, A344193, A344238, A344241, A344644.

N:= 10^5: # for terms <= N
V:= Vector(N, datatype=integer[4]):
for a from 1 while a^4 <= N do
  for b from 1 to a while a^4+b^4 <= N do
    for c from 1 to b while a^4 + b^4+ c^4 <= N do
      for d from 1 to c do
         v:= a^4+b^4+c^4+d^4;
         if v > N then break fi;
         V[v]:= V[v]+1
od od od od:
select(i -> V[i]>1, [$1..N]); # Robert Israel, Oct 07 2019

Select[Range@20000, Length@Select[PowersRepresentations[#, 4, 4], ! MemberQ[#, 0] &] > 1 &]

A025407 Numbers that are the sum of 4 positive cubes in 3 or more ways.

Original entry on oeis.org

1225, 1521, 1582, 1584, 1738, 1764, 1979, 2009, 2249, 2366, 2415, 2422, 2457, 2459, 2485, 2520, 2539, 2737, 2753, 2763, 2790, 2799, 3008, 3094, 3185, 3187, 3213, 3248, 3276, 3392, 3456, 3458, 3465, 3572, 3582, 3600, 3607, 3626, 3656, 3663, 3717, 3736

Offset: 1

David W. Wilson

nonn

Cf. A025368, A025398, A025405, A025406, A343704, A343971, A344241.

{n: A025457(n) >= 3}. - R. J. Mathar, Jun 15 2018

A344243 Numbers that are the sum of five fourth powers in three or more ways.

Original entry on oeis.org

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, 20995, 21235, 21250, 21330, 21364, 21410, 21954, 23139, 23795, 24754, 25810, 26578, 28610, 28930

Offset: 1

David Consiglio, Jr., May 12 2021

nonn

			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
so 6850 is a term of this sequence.

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

Cf. A342687, A343704, A344238, A344241, A344244, A344354, A345560.

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])

A344242 Numbers that are the sum of four fourth powers in exactly three ways.

Original entry on oeis.org

16578, 43234, 49329, 53218, 54978, 57154, 93393, 106354, 107649, 108754, 138258, 151219, 160434, 168963, 173539, 177699, 178738, 181138, 183603, 185298, 195378, 195859, 196418, 197154, 197778, 201683, 202419, 209763, 211249, 216594, 217138, 223074, 234274, 235554, 235569, 237249, 237699, 240834

Offset: 1

David Consiglio, Jr., May 12 2021

nonn

Differs from A344241 at term 36 because 236674 = 1^4 + 2^4 + 7^4 + 22^4 = 3^4 + 6^4 + 18^4 + 19^4 = 7^4 + 14^4 + 16^4 + 19^4 = 8^4 + 16^4 + 17^4 + 17^4

			49329 is a member of this sequence because 49329 = 2^4 + 2^4 + 12^4 + 13^4 = 4^4 + 8^4 + 9^4 + 14^4 = 6^4 + 9^4 + 12^4 + 12^4

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

Cf. A025405, A344193, A344240, A344241, A344244, A344353.

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,4):
    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])

A344352 Numbers that are the sum of four fourth powers in four or more 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

			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. A343971, A344241, A344277, A344353, A344354, A344356.

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])

A344239 Numbers that are the sum of three fourth powers in three or more ways.

Original entry on oeis.org

811538, 1733522, 2798978, 3750578, 4614722, 5978882, 6573938, 7303842, 8878898, 12771458, 12984608, 13760258, 14677362, 15601698, 15916082, 16196193, 17868242, 20621042, 21556178, 22349522, 22673378, 25190802, 25589858, 27736352, 29969282, 30623138, 33998258, 39765362, 41532498, 44048498

Offset: 1

David Consiglio, Jr., May 12 2021

nonn

			2798978 =  6^4 + 31^4 + 37^4
        =  9^4 + 29^4 + 38^4
        = 13^4 + 26^4 + 39^4
so 2798978 is a term of this sequence.

David Consiglio, Jr., Table of n, a(n) for n = 1..14213

Cf. A025398, A309762, A344240, A344241, A344277.

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,3):
    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])

A345337 Numbers that are the sum of four fifth powers in three or more ways.

Original entry on oeis.org

1479604544, 8429250269, 31738437018, 47347345408, 101802671905, 213625838382, 269736008608, 288202145792, 353946845525, 355891431456, 359543904192, 434029382875, 453675031150, 467943544849, 470899924000, 476304861791, 568433238331, 690221638656, 706199665600

Offset: 1

David Consiglio, Jr., Jun 14 2021

nonn

No numbers that are the sum of four fifth powers in four ways have been found. As a result, there is no corresponding sequence for the sum of four fifth powers in exactly three ways.

			8429250269 is a term because 8429250269 = 4^5 + 41^5 + 73^5 + 91^5  = 13^5 + 28^5 + 82^5 + 86^5  = 21^5 + 27^5 + 68^5 + 93^5.

Sean A. Irvine, Table of n, a(n) for n = 1..500

Cf. A342687, A344241, A344644.

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

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A309763 Numbers that are the sum of 4 nonzero 4th powers in more than one way.

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A025407 Numbers that are the sum of 4 positive cubes in 3 or more ways.

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A344243 Numbers that are the sum of five fourth powers in three or more ways.

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A344242 Numbers that are the sum of four fourth powers in exactly three ways.

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A344352 Numbers that are the sum of four fourth powers in four or more ways.

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A344239 Numbers that are the sum of three fourth powers in three or more ways.

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Python

A345337 Numbers that are the sum of four fifth powers in three or more ways.

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Python