cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A003339 Numbers that are the sum of 5 positive 4th powers.

Original entry on oeis.org

5, 20, 35, 50, 65, 80, 85, 100, 115, 130, 145, 165, 180, 195, 210, 245, 260, 275, 290, 305, 320, 325, 340, 355, 370, 385, 405, 420, 435, 450, 500, 515, 530, 545, 560, 580, 595, 610, 625, 629, 644, 659, 674, 675, 689, 690, 709, 724, 739, 754, 755, 770, 785, 789, 800
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 04 2020: (Start)
22418 is in the sequence as 22418 = 1^4 + 2^4 + 7^4 + 10^4 + 10^4.
30004 is in the sequence as 30004 = 2^4 + 3^4 + 5^4 + 11^4 + 11^4.
39028 is in the sequence as 39028 = 5^4 + 5^4 + 7^4 + 11^4 + 12^4. (End)

Links

Crossrefs

Supersequence of A047716.
Cf. A000583, A003328, A003338, A003340, A003350, A344190, A344238.

Programs

  • Mathematica
    Select[Range[1000], AnyTrue[PowersRepresentations[#, 5, 4], First[#]>0&]&] (* Jean-François Alcover, Jul 18 2017 *)
  • Python
    from itertools import combinations_with_replacement as combs_with_rep
def aupto(limit):
  qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 4 <= limit]
  ss = set(sum(c) for c in combs_with_rep(qd, 5))
  return sorted(s for s in ss if s <= limit)
print(aupto(800)) # Michael S. Branicky, May 20 2021

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

Views

Author

David Consiglio, Jr., May 17 2021

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Python
    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

Views

Author

Ilya Gutkovskiy, Aug 15 2019

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

A344237 Numbers that are the sum of five fourth powers in exactly two ways.

Original entry on oeis.org

260, 275, 340, 515, 884, 1555, 2595, 2660, 2675, 2690, 2705, 2755, 2770, 2835, 2930, 2945, 3010, 3185, 3299, 3314, 3379, 3554, 3923, 3970, 3985, 4050, 4115, 4145, 4160, 4210, 4290, 4355, 4400, 4465, 4594, 4769, 4834, 5075, 5090, 5155, 5265, 5330, 5395, 5440, 5505, 5570, 5699, 6370, 6545, 6580, 6595, 6660, 6675
Offset: 1

Views

Author

David Consiglio, Jr., May 12 2021

Keywords

Comments

Differs from A344237 at term 31 because 4225 = 2^4 + 2^4 + 2^4 + 3^4 + 8^4 = 2^4 + 4^4 + 4^4 + 6^4 + 7^4 = 3^4 + 4^4 + 6^4 + 6^4 + 6^4

Examples

			340 is a member of this sequence because 340 = 1^4 + 1^4 + 1^4 + 3^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4 + 3^4

Links

Crossrefs

Cf. A048927, A342686, A344190, A344193, A344238, A344244, A345814.

Programs

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

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

Views

Author

David Consiglio, Jr., May 12 2021

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Python
    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])

A345559 Numbers that are the sum of six fourth powers in two or more ways.

Original entry on oeis.org

261, 276, 291, 341, 356, 421, 516, 531, 596, 771, 885, 900, 965, 1140, 1361, 1509, 1556, 1571, 1636, 1811, 2180, 2596, 2611, 2661, 2676, 2691, 2706, 2721, 2741, 2756, 2771, 2786, 2836, 2851, 2916, 2931, 2946, 2961, 3011, 3026, 3091, 3186, 3201, 3220, 3266
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			276 is a term because 276 = 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 4^4 = 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4.

Links

Crossrefs

Cf. A003340, A344238, A345507, A345511, A345560, A345568, A345814.

Programs

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

A343702 Numbers that are the sum of five positive cubes in two or more ways.

Original entry on oeis.org

157, 220, 227, 246, 253, 260, 267, 279, 283, 286, 305, 316, 323, 342, 344, 361, 368, 377, 379, 384, 403, 410, 435, 440, 442, 468, 475, 487, 494, 501, 523, 530, 531, 549, 562, 568, 586, 592, 594, 595, 599, 602, 621, 625, 640, 647, 657, 658, 683, 703, 710, 712, 719, 729, 731, 738, 745, 752, 759, 764, 766, 771, 773, 778, 785
Offset: 1

Views

Author

David Consiglio, Jr., Apr 26 2021

Keywords

Comments

This sequence differs from A048927:
766 = 1^3 + 1^3 + 2^3 + 3^3 + 9^3
= 1^3 + 4^3 + 4^3 + 5^3 + 8^3
= 2^3 + 2^3 + 4^3 + 7^3 + 7^3.
So 766 is a term, but not a term of A048927.

Examples

			227 = 1^3 + 1^3 + 1^3 + 2^3 + 6^3
    = 2^3 + 3^3 + 4^3 + 4^3 + 4^3
so 227 is a term of this sequence.

Links

Crossrefs

Cf. A003328, A025406, A048927, A343704, A344238, A344795, A345511.

Programs

  • Mathematica
    Select[Range@1000,Length@Select[PowersRepresentations[#,5,3],FreeQ[#,0]&]>1&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)
  • Python
    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)]#n
for pos in cwr(power_terms,5):#m
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v >= 2])#s
for x in range(len(rets)):
    print(rets[x])
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.