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.

Previous Showing 21-23 of 23 results.

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 &]

A025367 Numbers that are the sum of 4 nonzero squares in 2 or more ways.

Original entry on oeis.org

28, 31, 34, 36, 37, 39, 42, 43, 45, 47, 49, 50, 52, 54, 55, 57, 58, 60, 61, 63, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A000414, A024796, A025358, A025368, A025406, A344795.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
V:= Vector(N):
for x from 1 while x^2 +3 <= N do
for y from 1 to x while x^2 + y^2 + 2 <= N do
  for z from 1 to y while x^2 + y^2 + z^2 + 1 <= N do
    for w from 1 to z while x^2 + y^2 + z^2 + w^2 <= N do
       t:= x^2 + y^2 + z^2 + w^2;
       V[t]:= V[t]+1;
od od od od:
select(t -> V[t] >= 2, [$1..N]); # Robert Israel, Jul 05 2017
  • Mathematica
    Select[Range@ 200, 2 == Length@ Quiet@ IntegerPartitions[#, {4}, Range[Sqrt@ #]^2, 2] &] (* Giovanni Resta, Jul 05 2017 *)
M = 1000;
Clear[V]; V[_] = 0;
For[a = 1, a <= Floor[Sqrt[M/4]], a++,
  For[b = a, b <= Floor[Sqrt[(M - a^2)/3]], b++,
    For[c = b, c <= Floor[Sqrt[(M - a^2 - b^2)/2]], c++,
      For[d = c, d <= Floor[Sqrt[M - a^2 - b^2 - c^2]], d++,
        m = a^2 + b^2 + c^2 + d^2;
        V[m] = V[m] + 1;
]]]];
Select[Range[M], V[#] >= 2&] (* Jean-François Alcover, Mar 22 2019, after Robert Israel *)

Formula

{n: A025428(n) >= 2}. - R. J. Mathar, Jun 15 2018

A282241 Numbers that are the sum of 3 distinct nonzero squares in two ways with symmetrical differences: a(n) = (p-a)^2+p^2+(p+b)^2 = (q-b)^2+q^2+(q+a)^2, p, q, a, b, positive integer, a

Original entry on oeis.org

62, 89, 101, 122, 134, 146, 150, 161, 173, 185, 189, 203, 206, 209, 218, 230, 234, 248, 254, 257, 266, 269, 270, 278, 281, 285, 299, 305, 314, 317, 321, 326, 329, 338, 341, 342, 347, 356, 357, 362, 374, 377, 378, 386, 389, 398, 401, 404, 405, 414, 419, 422, 425, 426, 434, 437, 441, 446, 449, 458
Offset: 1

Views

Author

Antonio Roldán, Feb 09 2017

Keywords

Comments

This sequence is subsequence of A004432 and A024804.
q-p is even, and b-a is multiple of 3, because 3(q-p)=2(b-a).

Examples

			122 = (5-1)^2+5^2+(5+4)^2 = (7-4)^2+7^2+(7+1)^2, with symmetrical differences 1 and 4.
248 = (6-2)^2+6^2+(6+8)^2 = (10-8)^2+10^2+(10+2)^2, with a=2, b=8.

Crossrefs

Cf. A004432, A024796, A024804.

Programs

  • PARI
    is_sym_sum(n)=local(x,e=0,a,b,p);x=1;while(x^2a,p=1;while(p^2<=n/3&&e==0,if(p^2+(p+b)^2+(p+a+b)^2==n,e=1);p+=1)));a+=1);x+=1);e
for(i=3,500,if(is_sym_sum(i),print1(i,", ")))
Previous Showing 21-23 of 23 results.

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