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 1-4 of 4 results.

A003348 Numbers that are the sum of 3 positive 5th powers.

Original entry on oeis.org

3, 34, 65, 96, 245, 276, 307, 487, 518, 729, 1026, 1057, 1088, 1268, 1299, 1510, 2049, 2080, 2291, 3072, 3127, 3158, 3189, 3369, 3400, 3611, 4150, 4181, 4392, 5173, 6251, 6282, 6493, 7274, 7778, 7809, 7840, 8020, 8051, 8262, 8801, 8832, 9043, 9375, 9824, 10902, 10933
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _David A. Corneth_, Aug 03 2020: (Start)
2656719 is in the sequence as 2656719 = 6^5 + 15^5 + 18^5.
8167555 is in the sequence as 8167555 = 14^5 + 19^5 + 22^5.
15646315 is in the sequence as 15646315 = 12^5 + 16^5 + 27^5. (End)

Links

Crossrefs

Cf. A000584, A003337, A003347, A003349, A344641, A345010.

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 &]

A344644 Numbers that are the sum of four fifth powers in two or more 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

Views

Author

David Consiglio, Jr., May 25 2021

Keywords

Examples

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

Links

Crossrefs

Cf. A003349, A309763, A342685, A344645, A345010, A345337.

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, 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])

A376289 Values k for primitive solutions to k^5 + a^5 + b^5 = c^5 + d^5 + e^5 with k >= a >= b >= 0 and k > c >= d >= e >= 0, repetitions allowed.

Original entry on oeis.org

66, 67, 74, 83, 107, 118, 119, 123, 136, 142, 152, 155, 169, 170, 181, 182, 186, 201, 204, 215, 216, 224, 229, 233, 234, 248, 258, 264, 274, 282, 283, 286, 288, 289, 293, 294, 307, 310, 310, 328, 331, 348, 364, 364, 373, 377, 378, 394, 399, 413, 417, 420, 421, 425, 426, 430, 430, 433, 436, 448, 459, 470, 474, 480, 486, 490, 498
Offset: 1

Views

Author

Artur Jasinski, Sep 19 2024

Keywords

Comments

This case is known in literature as 5.3.3 (see e.g. Eric Weisstein's World of Mathematics).
Primitive means a solution has gcd(k,a,b,c,d,e) = 1.
For primitive solutions of the 5.1.5 case see A063923.
For primitive solutions of the 5.2.4 case see A376914.
Although the definition does not require all coefficients to be nonzero or distinct, all known solutions have k > a > b > 0 and c > d > e > 0.
In every known case, k+a+b-c-d-e is even and very often zero.
This sequence is infinite as follows:
1) Bremner's modified one parameter identity (with conditions k+a+b-c-d-e=0 and k-a=c-d):
(37888 + 67978*w + 53683*w^2 + 24217*w^3 + 6750*w^4 + 1164*w^5 + 115*w^6 + 5*w^7)^5+
(15744 + 33046*w + 29861*w^2 + 15193*w^3 + 4738*w^4 + 912*w^5 + 101*w^6 + 5*w^7)^5+
(16376 + 33534*w + 29739*w^2 + 14937*w^3 + 4622*w^4 + 888*w^5 + 99*w^6 + 5*w^7)^5
=
(27912 + 52390*w + 43165*w^2 + 20281*w^3 + 5882*w^4 + 1056*w^5 + 109*w^6 + 5*w^7)^5+
(5768 + 17458*w + 19343*w^2 + 11257*w^3 + 3870*w^4 + 804*w^5 + 95*w^6 + 5*w^7)^5+
(36328 + 64710*w + 50775*w^2 + 22809*w^3 + 6358*w^4 + 1104*w^5 + 111*w^6 + 5*w^7)^5
which generate members of this sequence for nonnegative w=0,1,2,3,...
2) Moessner's one parameter identity (k+a+b-c-d-e=40*n)
(a^36 + 8*a^26 + 12*a^16 + 20*a^11 - a^6)^5+
(a^33 - 12*a^23 - 28*a^13 - a^3)^5+
(a^30 + 20*a^25 - 12*a^20 - 8*a^10 - 1)^5
=
(a^36 + 8*a^26 + 12*a^16 - 20*a^11 - a^6)^5+
(a^33 + 28*a^23 + 12*a^13 - a^3)^5+
(a^30 - 20*a^25 - 12*a^20 - 8*a^10 - 1)^5
which generate members of this sequence for a=2,3,4,...
3) Moessner's two parameter identity (with condition k+a+b-c-d-e=0):
(75*x^7-230*x^6*y-113*x^5*y^2+510*x^4*y^3-407*x^3*y^4+62*x^2*y^5+125*x*y^6-150*y^7)^5+
(-175*x^7+170*x^6*y-391*x^5*y^2-30*x^4*y^3+451*x^3*y^4-602*x^2*y^5+115*x*y^6-50*y^7)^5+
(175*x^7-160*x^6*y+387*x^5*y^2-108*x^4*y^3+5*x^3*y^4-336*x^2*y^5+265*x*y^6-100*y^7)^6
=
(25*x^7-290*x^6*y+689*x^5*y^2-138*x^4*y^3+27*x^3*y^4-62*x^2*y^5+155*x*y^6-150*y^7)^5+
(-25*x^7-653*x^5*y^2+564*x^4*y^3-195*x^3*y^4-208*x^2*y^5+105*x*y^6-100*y^7)^5+
(75*x^7+70*x^6*y-153*x^5*y^2-54*x^4*y^3+217*x^3*y^4-606*x^2*y^5+245*x*y^6-50*y^7)^5
4) Choudhry and Wróblewski two parameter identity:
(2 p^15 q + 6 p^5 q^11)^5 +
(p^16 - 3 p^11 q^5 - 5 p^6 q^10 - p q^15)^5 +
(6 p^11 q^5 + 2 p q^15)^5
= (p^16 + 3 p^11 q^5 - 5 p^6 q^10 + p q^15)^5 +
(p^15 q + 5 p^10 q^6 + 3 p^5 q^11 - q^16)^5 +
(p^15 q - 5 p^10 q^6 + 3 p^5 q^11 + q^16)^5
5) Edward Brisse two parameter identity (with condition k+a+b-c-d-e=0):
(2*a^8*b+10*a^7*b^2-20*a^6*b^3+20*a^5*b^4-34*a^4*b^5-10*a^3*b^6+270*a^2*b^7-20*a*b^8+682*b^9)^5+
(-2*a^8*b+10*a^7*b^2+20*a^6*b^3+20*a^5*b^4+34*a^4*b^5-10*a^3*b^6-270*a^2*b^7-20*a*b^8-682*b^9)^5+
(a^9-22*a^5*b^4-125*a^3*b^6-79*a*b^8)^5
=
(a^8*b+10*a^7*b^2-10*a^6*b^3+20*a^5*b^4-92*a^4*b^5-160*a^3*b^6-15*a^2*b^7-320*a*b^8+341*b^9)^5+
(-a^8*b+10*a^7*b^2+10*a^6*b^3+20*a^5*b^4+92*a^4*b^5-160*a^3*b^6+15*a^2*b^7-320*a*b^8-341*b^9)^5+
(a^9-22*a^5*b^4+175*a^3*b^6+521*a*b^8)^5
When we take b=1 in this identity we obtain the Lander 1968 one parameter identity.

Examples

			67^5 + 28^5 + 24^5 = 62^5 + 54^5 + 3^5 so 67 is a term.
399^5 + 237^5 + 62^5 = 382^5 + 307^5 + 9^5 so 399 is a term.
310^5 + 118^5 + 102^5 = 271^5 + 270^5 + 49^5 and 310^5 + 124^5 + 116^5 = 294^5 + 235^5 + 21^5 so 310 is a repeated term.

Links

Crossrefs

Cf. A063923, A345010, A376914.

Programs

  • Mathematica
    ww = {}; Monitor[Do[Do[Do[Do[kk = PowersRepresentations[e^5 + d^5 + c^5 - k^5, 2, 5];
If[kk != {},If[GCD[k, c, d, e, kk[[1]][[1]], kk[[1]][[2]]] == 1,
AppendTo[ww, k]; Print[k];Print[{k, kk[[1]][[2]], kk[[1]][[1]], c, d, e}]]], {e, 0, d}],{d, 0, c}], {c, 0, k - 1}], {k, 4, 186}], {c, k}];ww
  • PARI
    lista(maxk, prfull=0)={for(k=1, maxk, for(a=0, k, for(b=0, a, my(s=k^5+a^5+b^5); for(c=sqrtnint(s\3,5), k-1, for(d=sqrtnint((s-c^5-1)\2,5)+1, min(c, sqrtnint(s-c^5,5)), my(e); if(ispower(s-c^5-d^5,5,&e) && gcd([k,a,b,c,d,e])==1, if(prfull, print([k,a,b,c,d,e]), print1(k, ", ") )) )))))} \\ Andrew Howroyd, Oct 08 2024
Showing 1-4 of 4 results.

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

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