A004214 -id:A004214 - cp's OEIS frontend

A214329 Complement of A214328.

2, 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 109

Offset: 1

N. J. A. Sloane, Jul 26 2012, following a suggestion from Hans Isdahl, Apr 19 2012

nonn

Numbers that are the sum of 2 or 3 nonzero squares. - Altug Alkan, Jan 13 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000404, A000408, A004214, A018825, A214328.

is2(n) = {for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))}
is3(n) = {my(a, b) ; a=1; while(a^2+1Altug Alkan, Jan 13 2016

A267189 (Cubes of positive numbers) that are not the sum of three nonzero squares.

Original entry on oeis.org

1, 8, 64, 343, 512, 3375, 4096, 12167, 21952, 29791, 32768, 59319, 103823, 166375, 216000, 250047, 262144, 357911, 493039, 658503, 778688, 857375, 1092727, 1367631, 1404928, 1685159, 1906624, 2048383, 2097152, 2460375, 2924207, 3442951, 3796416, 4019679, 4657463, 5359375, 6128487

Offset: 1

N. J. A. Sloane, Jan 18 2016

nonn

This is the intersection of A004214 and A000578.

The original definition of A134739 was ambiguous - this is the second way it could have been interpreted. The other way, now the official definition of A134739, being "Cubes of (positive numbers that are not the sum of three nonzero squares)".

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000578, A004214, A134739.

is(n) = { my(a, b) ; a=1; while(a^2+1Altug Alkan, Jan 18 2016

More terms from Altug Alkan, Jan 18 2016

A125112 Numbers which are not the sum of 3 nonzero squares, but which can be expressed as the product of two numbers that are the sum of 3 nonzero squares.

Original entry on oeis.org

63, 87, 135, 156, 159, 183, 207, 231, 252, 279, 303, 319, 327, 348, 351, 375, 399, 423, 444, 447, 471, 476, 495, 519, 540, 543, 551, 567, 572, 583, 591, 615, 624, 636, 639, 663, 671, 687, 700, 711, 732, 735, 759, 783, 807, 828, 831, 847, 855, 879, 903, 924

Offset: 1

Artur Jasinski, Nov 21 2006

nonn

Intersection of A004214 with products of pairs of terms of A000408.

			a(2) = 87 = 3 * 29 = (1^2+1^2+1^2) * (4^2+3^2+2^2)
87 does not have a partition as a sum x^2+y^2+z^2 with x,y,z>0
63=3*21; 87=3*29; 135=3*45; 156=6*26; 572=22*26;

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000408 (sums of 3 nonzero squares), A004214 (not sums of 3 nonzero squares).

isA000408 := proc(n) local a,b,c2 ; a:=1; while a^2A125112 := proc(n) local d,i; if isA000408(n) then RETURN(false) ; else d := numtheory[divisors](n) ; for i from 1 to nops(d) do if isA000408(op(i,d)) and isA000408(n/op(i,d)) then RETURN(true) ; fi ; od ; RETURN(false) ; fi ; end: for an from 1 to 1600 do if isA125112(an) then printf("%d,",an) ; fi ; od ; # R. J. Mathar, Nov 23 2006

isA000408[n_] := Module[{a, b, c2}, a = 1; While[a^2 < n, b = 1; While[b <= a && a^2 + b^2 < n, c2 = n - a^2 - b^2; If[IntegerQ@Sqrt@c2, Return[True]]; b++]; a++]; Return[False]];
isA125112[n_] := Module[{d, i}, If[isA000408[n], Return[False], d = Divisors[n]; For[i = 1, i <= Length[d], i++, If[isA000408[d[[i]]] && isA000408[n/d[[i]]], Return[True]]]; Return[False]]];
Select[Range[1600], isA125112] (* Jean-François Alcover, Jul 22 2024, after R. J. Mathar *)

Edited and extended by R. J. Mathar and Ray Chandler, Nov 23 2006

A178615 Smaller of two consecutive numbers that are not the sum of 3 nonzero squares.

Original entry on oeis.org

1, 4, 7, 15, 31, 39, 63, 79, 111, 127, 159, 207, 231, 239, 255, 319, 367, 399, 447, 495, 511, 519, 591, 623, 639, 751, 831, 879, 927, 959, 1007, 1023, 1135, 1263, 1279, 1359, 1391, 1471, 1519, 1599, 1647, 1775, 1791, 1903, 1983, 2031, 2047, 2079, 2159, 2287

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 30 2010

nonn

{1,2,4,5,7,8,10,13,15,16,20,23,25,28,31,32,..}->{1,4,7,15,31,..}

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004214

q=66;q2=q^2+2;lst={};Do[Do[Do[z=a^2+b^2+c^2;If[z<=q2,AppendTo[lst,z]],{c,b,1,-1}],{b,a,1,-1}],{a,q}];lst; u=Union@lst;a=Complement[Range[q^2],u];lst={};Do[If[a[[n+1]]-a[[n]]==1,AppendTo[lst,a[[n]]]],{n,Length[a]-1}];lst

A180917 Numbers that are not the sum of three positive heptagonal numbers.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 38, 39, 40, 41, 44, 45, 46, 47, 49, 50, 51, 52, 55, 56, 58, 60, 61, 62, 64, 65, 66, 67, 68, 71, 72, 73, 76, 77, 78, 79, 81, 82, 84, 85, 87, 88, 92, 93, 94, 97, 98, 99, 101

Offset: 1

Jonathan Vos Post, Sep 23 2010

Complement of A117105. This is to heptagonal numbers A000566,
as A007536 is to hexagonal numbers A000384,
as A003679 is to pentagonal numbers A000326,
and as A004214 is to squares A000290.
This sequence is presumably finite: what is its likely last element?
Last element appears to be a(1671) = 273118. - Charles R Greathouse IV, Sep 27 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..1671

Cf. A000290, A000326, A000384, A000566, A003679, A004214, A007536, A117105.

A330708 Numbers that are not the sum of 2 nonzero squares and a positive cube.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 12, 15, 17, 20, 22, 23, 24, 31, 36, 39, 43, 50, 55, 57, 63, 65, 70, 71, 78, 87, 94, 103, 111, 113, 115, 119, 120, 134, 139, 141, 148, 160, 167, 169, 185, 204, 211, 254, 263, 267, 279, 283, 286, 302, 311, 312, 331, 335, 342, 349, 379, 391

Offset: 1

XU Pingya, Jun 08 2020

nonn

A022552 is a subsequence.

a(490) = A022552(434) = 5042631. No more terms <= 4 * 10^7.

Robert Israel, Table of n, a(n) for n = 1..490

Cf. A022552, A000534, A004214.

N:= 500: # for terms <= N
G1:= add(x^(i^2), i=1..floor(sqrt(N))):
G2:= add(x^(i^3), i=1..floor(N^(1/3))):
G:= expand(G1^2*G2):
select(t -> coeff(G,x,t)=0, [$0..N]); # Robert Israel, Jun 12 2020

m = 0;
n = 400.;
t = Union@Flatten@Table[x^2 + y^2 + z^3, {x, (n/2)^(1/2)}, {y, x, (n - x^2)^(1/2)}, {z, If[x^2 + y^2 < m, Floor[(m - 1 - x^2 - y^2)^(1/3)] + 1, 1], (n - x^2 - y^2)^(1/3)}];
Complement[Range[m, n], t]

A273123 Values of A007692(n) that are not of the form x^2 + y^2 + z^2 where x, y, z are nonzero integers.

Original entry on oeis.org

85, 130, 340, 520, 1360, 2080, 5440, 8320, 21760, 33280, 87040, 133120, 348160, 532480, 1392640, 2129920, 5570560, 8519680, 22282240, 34078720, 89128960, 136314880, 356515840, 545259520, 1426063360, 2181038080, 5704253440, 8724152320

Offset: 1

Altug Alkan, May 16 2016

nonn

If n is in this sequence, then 4*n is also in this sequence. So 85*4^k and 130*4^k are terms of this sequence for all nonnegative values of k.

For more details see A051952.

			85 is a term because 85 = 2^2 + 9^2 = 6^2 + 7^2 and 85 = x^2 + y^2 + z^2 has no solution for nonzero integer values of x, y, z.
130 is a term because 130 = 3^2 + 11^2 = 7^2 + 9^2 and 130 = x^2 + y^2 + z^2 has no solution for nonzero integer values of x, y, z.
340 is a term because 340 = 4*85 and 85 is a term.

Cf. A000408, A004214, A007692, A051952.

twoQ[n_] := 2 == Length@ Select[ PowersRepresentations[n, 2, 2], Times @@ # > 0 &, 2]; threeQ[n_] := {} != Quiet@ IntegerPartitions[n, {3}, Range[ Sqrt@ n]^2, 1]; Select[Range[10^5], twoQ[#] && ! threeQ[#] &] (* Giovanni Resta, May 16 2016 *)

a(14)-a(28) from Giovanni Resta, May 16 2016

A335658 Numbers that are not the sum of 3 nonzero squares and a positive cube.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 16, 21, 24, 40

Offset: 1

XU Pingya, Jun 17 2020

No more terms up to 6*10^5.

Cf. A000534, A004214, A297788.

n = 5 * 10^4.;
t = Union@Flatten@Table[x^2 + y^2 + z^2 + w^3, {x,(n/3)^(1/2)}, {y,x,((n-x^2)/2)^(1/2)},{z,y,(n-x^2-y^2)^(1/2)}, {w,(n-x^2-y^2-z^2)^(1/3)}];
Complement[Range[0,n],t]

A335659 Numbers that are not the sum of 3 nonzero squares and a positive 5th power.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 11, 14, 16, 17, 21, 24, 26, 29, 32, 33, 40, 48, 64, 72, 96, 112, 144, 160, 192, 240

Offset: 1

XU Pingya, Jun 17 2020

No more terms up to 10^6.

Cf. A000534, A004214, A273915.

n = 10^5;
t = Union@Flatten@Table[x^2 + y^2 + z^2 + w^5, {x, (n/3)^(1/2)}, {y, x, ((n-x^2)/2)^(1/2)}, {z, y, (n-x^2-y^2)^(1/2)}, {w, 1, (n-x^2-y^2-z^2)^(1/5)}];
Complement[Range[0, n], t]

cp's OEIS Frontend

A214329 Complement of A214328.

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A267189 (Cubes of positive numbers) that are not the sum of three nonzero squares.

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A125112 Numbers which are not the sum of 3 nonzero squares, but which can be expressed as the product of two numbers that are the sum of 3 nonzero squares.

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A178615 Smaller of two consecutive numbers that are not the sum of 3 nonzero squares.

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A180917 Numbers that are not the sum of three positive heptagonal numbers.

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A330708 Numbers that are not the sum of 2 nonzero squares and a positive cube.

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A273123 Values of A007692(n) that are not of the form x^2 + y^2 + z^2 where x, y, z are nonzero integers.

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A335658 Numbers that are not the sum of 3 nonzero squares and a positive cube.

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A335659 Numbers that are not the sum of 3 nonzero squares and a positive 5th power.

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