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 1-5 of 5 results.

A025398 Numbers that are the sum of 3 positive cubes in 3 or more ways.

Original entry on oeis.org

5104, 9729, 12104, 12221, 12384, 13896, 14175, 17604, 17928, 19034, 20691, 21412, 21888, 24480, 28792, 29457, 30528, 31221, 32850, 34497, 35216, 36288, 38259, 39339, 39376, 40041, 40060, 40097, 40832, 40851, 41033, 41040, 41364, 41966, 42056, 42687
Offset: 1

Views

Author

David W. Wilson

Keywords

Examples

			a(1) = A230477(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3. - _Jonathan Sondow_, Oct 24 2013

Links

Crossrefs

Cf. A008917, A018787, A025331, A025397, A025402, A025407, A230477, A343968, A344239.

Programs

  • Mathematica
    Select[ Range[ 50000], 2 < Length @ Cases[ PowersRepresentations[#, 3, 3], {?Positive, ?Positive, A008917%20by%20_Jonathan%20Sondow">?Positive}] &] (* adapted from Alcover's program for A008917 by _Jonathan Sondow, Oct 24 2013 *)
  • PARI
    is(n)=k=ceil((n-2)^(1/3)); d=0; for(a=1,k,for(b=a,k,for(c=b,k,if(a^3+b^3+c^3==n,d++))));d
n=3;while(n<50000,if(is(n)>=3,print1(n,", "));n++) \\ Derek Orr, Aug 27 2015

Formula

A008917(n) < a(n) <= A025397(n). - Jonathan Sondow, Oct 24 2013
{n: A025456(n) >= 3}. - R. J. Mathar, Jun 15 2018

A309762 Numbers that are the sum of 3 nonzero 4th powers in more than one way.

Original entry on oeis.org

2673, 6578, 16562, 28593, 35378, 42768, 43218, 54977, 94178, 105248, 106353, 122018, 134162, 137633, 149058, 171138, 177042, 178737, 181202, 195122, 195858, 198497, 216513, 234273, 235298, 235553, 264113, 264992, 300833, 318402, 318882, 324818, 334802, 346673
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 15 2019

Keywords

Examples

			2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4, so 2673 is in the sequence.

Links

Crossrefs

Cf. A003337, A008917, A018786, A024796, A193243, A309763, A344192, A344239, A345010.

Programs

  • Maple
    N:= 10^6: # 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 do
      v:= a^4+b^4+c^4;
      if v > N then break fi;
      V[v]:= V[v]+1
od od od:
select(i -> V[i]>1, [$1..N]); # Robert Israel, Aug 19 2019
  • Mathematica
    Select[Range@350000, Length@Select[PowersRepresentations[#, 3, 4], ! MemberQ[#, 0] &] > 1 &]

A344241 Numbers that are the sum of four fourth powers in three or more 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, 236674, 237249, 237699
Offset: 1

Views

Author

David Consiglio, Jr., May 12 2021

Keywords

Examples

			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
so 49329 is a term of this sequence.

Links

Crossrefs

Cf. A025407, A309763, A344239, A344242, A344243, A344352, A345337.

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,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])

A344240 Numbers that are the sum of three fourth powers in exactly three ways.

Original entry on oeis.org

811538, 1733522, 2798978, 3750578, 4614722, 6573938, 7303842, 8878898, 12771458, 12984608, 13760258, 14677362, 15601698, 16196193, 17868242, 21556178, 22349522, 25190802, 25589858, 27736352, 29969282, 41532498, 44048498, 44783648, 45182018, 50944418, 54894242, 57052562, 59165442, 60009248
Offset: 1

Views

Author

David Consiglio, Jr., May 12 2021

Keywords

Comments

Differs from A344239 at term 6 because 5978882 = 3^4 + 40^4 + 43^4 = 8^4 + 37^4 + 45^4 = 15^4 + 32^4 + 47^4 = 23^4 + 25^4 + 48^4

Examples

			2798978 is a member of this sequence because 2798978 = 6^4 + 31^4 + 37^4 = 9^4 + 29^4 + 38^4 = 13^4 + 26^4 + 39^4

Links

Crossrefs

Cf. A025397, A344192, A344239, A344242, A344278.

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,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])

A344277 Numbers that are the sum of three fourth powers in four or more ways.

Original entry on oeis.org

5978882, 15916082, 20621042, 22673378, 30623138, 33998258, 39765362, 48432482, 53809938, 61627202, 65413922, 74346818, 84942578, 88258898, 95662112, 103363442, 117259298, 128929682, 131641538, 137149922, 143244738, 155831858, 158811842, 167042642, 174135122, 175706258, 188529362
Offset: 1

Views

Author

David Consiglio, Jr., May 13 2021

Keywords

Examples

			20621042 is a member of this sequence because 20621042 = 5^4 + 54^4 + 59^4 = 10^4 + 51^4 + 61^4 = 25^4 + 46^4 + 63^4 = 26^4 + 39^4 + 65^4

Links

Crossrefs

Cf. A343968, A344239, A344278, A344352, A344364.

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,3):
    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])
Showing 1-5 of 5 results.

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