A344644 -id:A344644 - cp's OEIS frontend

A003349 Numbers that are the sum of 4 positive 5th powers.

4, 35, 66, 97, 128, 246, 277, 308, 339, 488, 519, 550, 730, 761, 972, 1027, 1058, 1089, 1120, 1269, 1300, 1331, 1511, 1542, 1753, 2050, 2081, 2112, 2292, 2323, 2534, 3073, 3104, 3128, 3159, 3190, 3221, 3315, 3370, 3401, 3432, 3612, 3643, 3854, 4096, 4151, 4182, 4213

Offset: 1

N. J. A. Sloane

			From _David A. Corneth_, Aug 03 2020: (Start)
670593 is in the sequence as 670593 = 1^5 + 8^5 + 10^5 + 14^5.
862512 is in the sequence as 862512 = 7^5 + 9^5 + 12^5 + 14^5.
1892695 is in the sequence as 1892695 = 1^5 + 1^5 + 5^5 + 18^5. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000584, A003338, A003348, A003350, A344642, A344644.

f@n_:= Select[Range@n,IntegerPartitions[#,{4},Range@(n^(1/5))^5] != {} &]; f@10000 (* Hans Rudolf Widmer, Dec 04 2022 *)

Incorrect program removed by David A. Corneth, Aug 03 2020

A342685 Numbers that are the sum of five fifth powers in two or more ways.

Original entry on oeis.org

4097, 51446, 51477, 51688, 52469, 54570, 59221, 68252, 68905, 84213, 110494, 131104, 151445, 212496, 300277, 325174, 325713, 355114, 422135, 422738, 589269, 637418, 794434, 810820, 876734, 876765, 876976, 877757, 879858, 884509, 893540, 909501, 924912, 935782, 976733, 995571, 1037784, 1083457

Offset: 1

David Consiglio, Jr., May 17 2021

nonn

			51477 = 2^5 + 4^5 + 7^5 + 7^5 + 7^5
      = 2^5 + 5^5 + 6^5 + 6^5 + 8^5
so 51477 is a term.

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

Cf. A003350, A342686, A342687, A344238, A344644, A345507.

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 >= 2])
for x in range(len(rets)):
    print(rets[x])

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

Original entry on oeis.org

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 &]

A344645 Numbers that are the sum of four fifth powers in exactly two ways.

Original entry on oeis.org

51445, 876733, 1646240, 3558289, 4062500, 5687000, 7962869, 8227494, 9792364, 9924675, 10908544, 12501135, 15249850, 18317994, 18804544, 20611151, 20983875, 21297837, 23944908, 24201342, 24598407, 27806867, 28055456, 29480343, 31584102, 32557875, 32814683, 35469555, 40882844, 45177175

Offset: 1

David Consiglio, Jr., May 25 2021

nonn

Differs from A344644 at term 508 because 1479604544 = 3^5 + 49^5 + 53^5 + 62^5 = 14^5 + 37^5 + 52^5 + 65^5 = 19^5 + 37^5 + 45^5 + 67^5

			1646240 is a term because 1646240 = 9^5 + 15^5 + 15^5 + 15^5 = 11^5 + 13^5 + 13^5 + 17^5

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

Cf. A342686, A344193, A344642, 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, 500)]
for pos in cwr(power_terms, 4):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k, v in keep.items() if v == 2])
for x in range(len(rets)):
    print(rets[x])

A345010 Numbers that are the sum of three fifth powers in two or more ways.

Original entry on oeis.org

1375298099, 1419138368, 2370099168, 5839897526, 16681039431, 27326512069, 28461637018, 34090335168, 44009539168, 45412427776, 47166830151, 57788232400, 75843173376, 89516861675, 89636142881, 140201053499, 186876720832, 191701358025, 209797492893, 220333644849

Offset: 1

David Consiglio, Jr., Jun 14 2021

nonn

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

			1419138368 is a term because 1419138368 = 13^5 + 51^5 + 64^5  = 18^5 + 44^5 + 66^5.

David Consiglio, Jr., Table of n, a(n) for n = 1..396

Cf. A003348, A309762, 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, 3):
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k, v in keep.items() if v >= 2])
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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A003349 Numbers that are the sum of 4 positive 5th powers.

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A342685 Numbers that are the sum of five fifth powers in two or more ways.

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

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A344645 Numbers that are the sum of four fifth powers in exactly two ways.

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A345010 Numbers that are the sum of three fifth powers in two or more ways.

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

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