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 21-30 of 30 results.

A004440 Numbers that are not the sum of 3 distinct nonzero squares.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 31, 32, 33, 34, 36, 37, 39, 40, 43, 44, 47, 48, 51, 52, 55, 57, 58, 60, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 82, 85, 87, 88, 92, 95, 96, 97, 99, 100, 102, 103, 108, 111, 112, 119, 123, 124, 127, 128, 130, 132, 135, 136, 143, 144, 148, 151, 156, 159, 160, 163, 167, 172, 175, 176
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A004432 (complement).

Programs

  • Maple
    N:= 1000: # to get all terms <= N
V:= Vector(N):
for x from 1 to floor(sqrt(N/3)) do
  for y from x+1 to floor(sqrt((N-x^2)/2)) do
    zs:= [$(y+1).. floor(sqrt(N-x^2-y^2))];
    V[map(z -> x^2 + y^2 + z^2, zs)]:= 1;
  od
od:
select(i -> V[i] = 0, [$1..N]); # Robert Israel, Dec 31 2015

A025340 Numbers that are the sum of 3 distinct nonzero squares in exactly 2 ways.

Original entry on oeis.org

62, 69, 74, 77, 86, 89, 90, 94, 98, 105, 117, 122, 125, 129, 131, 138, 141, 150, 154, 155, 158, 165, 166, 170, 179, 181, 195, 197, 201, 203, 210, 213, 217, 218, 225, 227, 229, 233, 238, 241, 242, 246, 248, 249, 250, 259, 273, 274, 275, 276, 282, 296, 297, 301, 308, 310
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Subsequence of A004432. Subsequence of A024804.

Programs

  • Maple
    N:= 10^6;
A:= Vector(N):
for a from 1 to floor(sqrt(N/3)) do
  for b from a+1 to floor(sqrt((N-a^2)/2)) do
    c:= [$(b+1) .. floor(sqrt(N-a^2-b^2))]:
    v:= map(t -> a^2 + b^2 + t^2, c):
    A[v]:= map(`+`,A[v],1)
od od:
select(t -> A[t]=2,[$1..N]); # Robert Israel, Jan 03 2016
  • Mathematica
    upperbound = 10^4; max = Floor@Sqrt@upperbound;
range = ConstantArray[0, 3*max^2];
++range[[#]]&/@(Plus@@#&/@Subsets[Range@max^2,{3}]);
Select[Flatten@Position[range, 2], # <= upperbound &] (* Hans Rudolf Widmer, Aug 04 2021 *)

A236300 Numbers n of the form x^3 + y^3 + z^3 - 3*x*y*z for x,y,z >= 0, where x + y + z < n.

Original entry on oeis.org

8, 9, 16, 18, 20, 27, 28, 32, 35, 36, 40, 44, 45, 49, 52, 54, 56, 63, 64, 65, 68, 70, 72, 76, 77, 80, 81, 88, 90, 91, 92, 98, 99, 100, 104, 108, 112, 116, 117, 119, 124, 125, 126, 128, 130, 133, 135, 136, 140, 143, 144, 148, 152, 153, 154, 160, 161, 162, 164, 169
Offset: 1

Views

Author

Arkadiusz Wesolowski, Apr 21 2014

Keywords

Comments

x^3 + y^3 + z^3 - 3*x*y*z = (x + y + z)*(x^2 + y^2 + z^2 - x*y - x*z - y*z), hence all terms are composite.
From Wolfdieter Lang, Apr 30 2014: (Start)
Take x >= y >= z >= 0, not all identical: the numbers are of the form (x + y + z)*(u^2 + v^2 + w^2)/2, where u = x-y, v = x-z, w = y-z, with u >= 0, v >=0, w >= 0, u - v + w = 0 and u^2 + v^2 + w^2 >= 4.
(i) If, say, x = y but not equal to z, then the numbers are of the form (2*x+y)*(x-z)^2 with x-z >= 2, z >= 0. Similarly for the other case with y = z not equal to x.
(ii) If x, y and z are distinct, u >= 1, v >= 1 and w >= 1, hence u is not equal to v, and v is not equal to w (because u - v + w = 0). (iia) If u = w then the numbers are of the form 3*y*3*(y-z)^2 with y-z >= 1, z >= 0. (iib) If the u, v, w are distinct >= 1 then the even members of the sequence A004432 with multiplicities A025442 are of interest. But only those (u, v, w) qualify which satisfy u - v + w = 0. E.g., A025442(5) = 30 = 1^2 + 2^2 + 5^2 does not qualify because no permutation of 1, 2, 5 works for u, v, w. A025442(1) = 14 qualifies because (u, v, w) = (2, 3, 1) satisfies 2 - 3 + 1 = 0. Then [x, y, z] = [4, 2, 1] and the number is 7*14/2 = 49.
(End)
The even numbers qualifying for the case (iib) above are shown in A240227 with the multiplicities A240228. - Wolfdieter Lang, May 02 2014

Examples

			From _Wolfdieter Lang_, Apr 30 2014: (Start)
The numbers of type (i) are seq((2*x+z)*(z-x)^2, z=0..(x-2)) (if x >= 2) and seq((2*x+z)*(z-x)^2, z >= (x+2)) for x = 0, 1, 2, ... E.g., x = 3:  54, 28, 44, and 108, 208, 350, 540, 784, 1088, 1458, 1900, 2420, 3024, ...
The numbers of type (iia) are [seq(9*y*(y-z)^2, y >= 1+z)] for z = 0, 1, 2, ... E.g., z=3: 36, 180, 486, 1008, 1800, 2916, 4410, ...
The numbers of type (iib) come from the even members 14, 26, 30, 38, 42, 46, 50, ... of A025442 (each with multiplicity 1) except of 30 (as explained above in a comment), 46 with 1, 3, 6 which is out, and also 50 with 3, 4, 5.  7*14/2 = 49 (see the comment above); 10*26/2 = 130 from (u, v, w) = (1, 4, 3) and [x, y, z] = [5, 4, 1]; 11*38/2 = 209 from (2, 5, 3) and [6, 4, 1]; 12*42/2 = 252 from (1, 5, 4) and [6, 5, 1]; ...
(End)

Crossrefs

Subsequence of A002808 (the composite numbers). A004432, A025442.

A242675 Smallest prime with exactly n representations as sum of 3 distinct positive squares.

Original entry on oeis.org

2, 29, 89, 101, 281, 269, 641, 461, 701, 761, 1049, 941, 1109, 1601, 1361, 2621, 2309, 1889, 2441, 2141, 2609, 3929, 3701, 3461, 3449, 5849, 4241, 4289, 5081, 8669, 7589, 5381, 6569, 9941, 8861, 7229, 7829, 8501, 8069, 8609, 9749, 10601
Offset: 0

Views

Author

Zak Seidov, May 20 2014

Keywords

Comments

2 cannot be represented as the sum of 3 distinct positive squares hence a(0)=2 (and offset is 0).

Examples

			29 = 2^2 + 3^2 + 4^2 and this is the only such representation.
89 = 2^2 + 6^2 + 7^2 = 3^2 + 4^2 + 8^2 and these are the only such representations.
101 = 1^2 + 6^2 + 8^2 = 2^2 + 4^2 + 9^2 = 4^2 + 6^2 + 7^2 and these are the only such representations.

Links

Crossrefs

Cf. A004432, A125516.

A267983 Integers n such that n^3 = (x^2 + y^2 + z^2) / 3 where x > y > z > 0, is soluble.

Original entry on oeis.org

3, 6, 7, 9, 10, 11, 12, 14, 15, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 86, 87, 88, 89, 90, 91, 92, 94
Offset: 1

Views

Author

Altug Alkan, Jan 23 2016

Keywords

Comments

Motivation was this simple question: What are the cubes that are the averages of 3 nonzero distinct squares?
Corresponding cubes are 27, 216, 343, 729, 1000, 1331, 1728, 2744, 3375, 4913, 5832, 6859, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 27000, ...
Complement of this sequence for positive integers is 1, 2, 4, 5, 8, 13, 16, 20, 21, 29, 32, 37, 45, 52, 53, 61, 64, 69, 77, ...
The positive cubes that are not the averages of 3 nonzero distinct squares are 1, 8, 64, 125, 512, 2197, 4096, 8000, 9261, 24389, 32768, 50653, 91125, ...

Examples

			3 is a term since 3^3 is the average of 1^2, 4^2, 8^2. 3^3 = (1^2 + 4^2 + 8^2) / 3.

Crossrefs

Cf. A004432, A117642.

Programs

  • Mathematica
    Select[Range@ 94, Resolve[Exists[{x, y, z}, Reduce[#^3 == (x^2 + y^2 + z^2)/3, {x, y, z}, Integers], x > y > z > 0]] &] (* Michael De Vlieger, Jan 24 2016 *)
  • PARI
    isA004432(n) = for(x=1, sqrtint(n\3), for(y=x+1, sqrtint((n-1-x^2)\2), issquare(n-x^2-y^2) && return(1)));
for(n=1, 1e2, if(isA004432(3*n^3), print1(n, ", ")));

A343099 Sums of 3 distinct odd squares.

Original entry on oeis.org

35, 59, 75, 83, 91, 107, 115, 131, 139, 147, 155, 171, 179, 195, 203, 211, 219, 227, 235, 243, 251, 259, 275, 283, 291, 299, 307, 315, 323, 331, 339, 347, 355, 363, 371, 379, 387, 395, 403, 411, 419, 427, 435, 443, 451, 459, 467, 475, 483, 491, 499, 507, 515, 523, 531
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 05 2021

Keywords

Comments

From Robert Israel, Apr 06 2021: (Start)
All terms == 3 (mod 8).
Conjecture: contains all numbers == 3 (mod 8) except 3, 11, 19, 27, 43, 51, 67, 99, 123, 163, 187, 267, 627. (End)

Examples

			107 is in the sequence since 107 = 1^2 + 5^2 + 9^2.

Links

Crossrefs

Subsequence of A017101.
Cf. A004432, A016754 (odd squares).

Programs

  • Maple
    N:= 10^4: # for terms <= N
S:= {seq(seq(seq(x^2+y^2+z^2, z = 1 .. min(y-2, floor(sqrt(N-x^2-y^2))), 2),y = 1 .. min(x-2, floor(sqrt(N-x^2))), 2), x = 1 .. floor(sqrt(N)),2)}:
sort(convert(S,list)); # Robert Israel, Apr 05 2021

A162163 Primes p such that p-1 and p+1 can individually be written as a sum of 2 and also as a sum of 3 distinct nonzero squares.

Original entry on oeis.org

179, 467, 739, 809, 1097, 1171, 1619, 1801, 1873, 1907, 2467, 3203, 3331, 3491, 3923, 4051, 4177, 4211, 4931, 5507, 5651, 6067, 6121, 6353, 6569, 6659, 7219, 8081, 8243, 8297, 8353, 8819, 9091, 9161, 9377, 10243, 10531, 10657, 10729, 10889, 11251, 11699
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jun 26 2009, Jun 27 2009

Keywords

Comments

A subsequence of A162164.

Examples

			p=12113: p-1=12112 = 36^2+40^2+96^2 = 36^2+104^2; p+1=12114 = 33^2+63^2+84^2 = 33^2+105^2.
p=4177: p-1=4176 = 24^2+60^2 = 24^2+36^2+48^2; p+1=4178 = 37^2+53^2 = 37^2+28^2+45^2. - _Vladimir Joseph Stephan Orlovsky_, Jun 26 2009
p=179: p-1=178 = 3^2+13^2 = 3^2+5^2+12^2; p+1=180 = 6^2+12^2=4^2+8^2+10^2. - _R. J. Mathar_, Jul 02 2009

Programs

  • Maple
    isA004431 := proc(n) local x,y ; for x from 1 do if x^2 > n then RETURN(false); fi; y := n-x^2 ; if y> 0 and issqr(y ) then y := sqrt(y) ; if y <> x then RETURN(true) ; fi; fi; od: end:
isA004432 := proc(n) local x,y,z ; for x from 1 do if x^2 > n then RETURN(false); fi; for y from x+ 1 do if x^2+y^2>n then break ; fi; z := n-x^2-y^2 ; if z> 0 and issqr(z ) then z := sqrt(z) ; if z > y and z > x then RETURN(true) ; fi; fi; od: od: end:
for n from 1 to 2000 do p := ithprime(n) ; if isA004432(p-1) and isA004432(p+1) and isA004431(p-1) and isA004431(p+1) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jul 02 2009
  • Mathematica
    f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++; If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[p=Prime[n];x=p-1;y=p+1;If[f[x]> 0&&f[y]>0,a=x-(f[x])^2;b=y-(f[y])^2;If[f[a]>0&&f[b]>0,c=(x-(f[x])^2-(f[a])^2)^(1/ 2);d=(y-(f[y])^2-(f[b])^2)^(1/2);If[c!=f[x]&&c!=f[a]&&f[x]!=f[a], If[d!=f[y]&&d!=f[b]&&f[y]!=f[b],AppendTo[lst,p]]]]],{n,3,6*6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 26 2009 *)

Formula

{p=A000040(i): p-1 in A004431 and p-1 in A004432 and p+1 in A004431 and p+1 in A004432}. - R. J. Mathar, Jul 02 2009

Extensions

Definition corrected, Mathematica duplicate removed, missing values added by R. J. Mathar, Jul 02 2009

A267986 Perfect powers of the form x^2 + y^2 + z^2 where x > y > z > 0.

Original entry on oeis.org

49, 81, 121, 125, 169, 196, 216, 225, 243, 289, 324, 361, 441, 484, 529, 625, 676, 729, 784, 841, 900, 961, 1000, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2187, 2197, 2209, 2401, 2500, 2601, 2704, 2744, 2809, 2916, 3025, 3125, 3136
Offset: 1

Views

Author

Altug Alkan, Jan 23 2016

Keywords

Comments

Intersection of A001597 and A004432.
Note that this sequence is not the complement of A267321. This sequence is a subsequence for complement of A267321.
Sequence focuses on the equation m^k = x^2 + y^2 + z^2 where x > y > z > 0 and m > 0, k >= 2.
Corresponding exponents are 2, 4, 2, 3, 2, 2, 3, 2, 5, 2, 2, 2, 2, 2, 2, 4, 2, 6, 2, 2, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 3, 2, 4, 2, 2, ...

Examples

			49 is a term because 49 = 7^2 = 2^2 + 3^2 + 6^2.
81 is a term because 81 = 9^2 = 1^2 + 4^2 + 8^2.
121 is a term because 121 = 11^2 = 2^2 + 6^2 + 9^2.

Crossrefs

Cf. A001597, A004432, A266927, A267321.

Programs

  • Mathematica
    fQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Range@ 1800, fQ@ # && Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]] &] (* Michael De Vlieger, Jan 24 2016, after Ant King at A001597 *)
  • PARI
    isA004432(n) = for(x=1, sqrtint(n\3), for(y=x+1, sqrtint((n-1-x^2)\2), issquare(n-x^2-y^2) && return(1)));
for(n=1, 1e4, if(isA004432(n) && ispower(n), print1(n, ", ")));

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,", ")))

A337759 Squares that are the sum of 3 distinct nonzero squares.

Original entry on oeis.org

49, 81, 121, 169, 196, 225, 289, 324, 361, 441, 484, 529, 625, 676, 729, 784, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3600, 3721, 3844, 3969
Offset: 1

Views

Author

Joseph Caliendo, Sep 18 2020

Keywords

Comments

These are the squares in A004432. - Omar E. Pol, Sep 18 2020

Examples

			49 is a term because 6^2(36) + 3^2(9) + 2^2(4) = 7^2(49).
81 is a term because 8^2(64) + 4^2(16) + 1^2(1) = 9^2(81).
121 is a term because 9^2(81) + 6^2(36) + 2^2(4) = 11^2(121).
625 is a term because 9^2(81) + 12^2(144) + 20^2(400) = 25^2(625).

Crossrefs

Cf. A004432, A161992.

Programs

  • Mathematica
    Select[Range[63]^2, Length @ Reduce[x^2 + y^2 + z^2 == # && 0 < x < y < z, {x, y, z}, Integers] > 0 &] (* Amiram Eldar, Sep 18 2020 *)

Formula

a(n) = A161992(n)^2. - Andrew Howroyd, Sep 18 2020
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