A022552 -id:A022552 - cp's OEIS frontend

A022551 Numbers that are the sum of 2 squares and a nonnegative cube.

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62

Offset: 1

N. J. A. Sloane

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..943
Index entries for sequences related to sums of squares

Complement of A022552.

isA022551 := proc(n)
    local a,b,c ;
    for c from 0 do
        if c^3 > n then
            return false;
        end if;
        for a from 0 do
            b2 := n-c^3-a^2 ;
            if b2 < a^2 then
                break;
            end if;
            if issqr(b2) then
                return true;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, Sep 02 2016

max = 1000;
Table[x^2 + y^2 + z^3, {x, 0, Sqrt[max]}, {y, x, Sqrt[max - x^2]}, {z, 0, (max - x^2 - y^2)^(1/3)}] // Flatten // Union // Select[#, # <= max&]& (* Jean-François Alcover, Mar 23 2020 *)

A115160 Numbers that are not the sum of two triangular numbers and a fourth power.

Original entry on oeis.org

33, 63, 75, 125, 365, 489, 492, 684, 693, 723, 954, 1043, 1185, 1505, 1623, 1629, 1736, 1775, 1899, 1904, 1925, 2015, 2051, 2679, 2883, 3534, 3774, 3936, 4332, 4461, 4739, 4923, 5445, 5721, 5847, 6285, 6348, 6474, 6783, 7034, 7478, 8604, 9576, 9686, 9863

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

There are 88 such numbers up to 2*10^9, the last one in this range being 1945428.

David W. Wilson, Table of n, a(n) for n = 1..88

Cf. A022552, A014158, A115159, A115161, A115162, A115163.

sumset_lim(a,b,lim)=my(v=[],u,t);if(a==b,for(i=1,#a,u=List();for(j=i,#b,t=a[i]+b[j];if(t>lim,break);listput(u,t));v=vecsort(concat(v,Vec(u)),,8)),for(i=1,#a,u=List();for(j=1,#b,t=a[i]+b[j];if(t>lim,break);listput(u,t));v=vecsort(concat(v,Vec(u)),,8)));v
makev(lim)=my(n=floor(sqrt(2*lim)-1/2),v);sumset_lim(v=vector(n,k,k*(k-1)/2),v,lim)
is(n)=for(i=1,#v,if(ispower(n-v[i],4),return(0));if(v[i]>n,return(1)))
v=makev(1e5);
for(n=1,1e5,if(is(n),print1(n", "))) \\ Charles R Greathouse IV, Aug 17 2011

A274274 Number of ordered ways to write n as x^3 + y^2 + z^2, where x,y,z are nonnegative integers with y <= z.

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 1, 0, 2, 3, 3, 1, 1, 2, 1, 0, 2, 3, 3, 1, 1, 2, 0, 0, 1, 3, 4, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 4, 1, 0, 3, 2, 2, 1, 2, 3, 1, 1, 1, 2, 3, 2, 3, 4, 1, 0, 1, 1, 3, 2, 1, 3, 1, 1, 3, 4, 4, 1, 3, 3, 0, 0, 4, 5, 3, 1, 2, 3, 0, 1, 4

Offset: 0

Zhi-Wei Sun, Jul 14 2016

nonn

Conjecture: Let n be any nonnegative integer.
(i) Either a(n) > 0 or a(n-2) > 0. Also, a(n) > 0 or a(n-6) > 0. Moreover, if n has the form 2^k*(4m+1) with k and m nonnegative integers, then a(n) > 0 except for n = 813, 4404, 6420, 28804.
(ii) Either n or n-3 can be written as x^3 + y^2 + 3*z^2 with x,y,z nonnegative integers.
(iii) For each d = 4, 5, 11, 12, either n or n-d can be written as x^3 + y^2 + 2*z^2 with x,y,z nonnegative integers.
We have verified that a(n) or a(n-2) is positive for every n = 0..2*10^6. Note that for each n = 0,1,2,... either n or n-2 can be written as x^2 + y^2 + z^2 with x,y,z nonnegative integers, which follows immediately from the Gauss-Legendre theorem on sums of three squares.

			a(6) = 1 since 6 = 1^3 + 1^2 + 2^2.
a(14) = 1 since 14 = 1^3 + 2^2 + 3^2.
a(31) = 1 since 31 = 3^3 + 0^2 + 2^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000290, A000578, A022551, A022552, A262857, A272979.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^3-y^2],r=r+1],{x,0,n^(1/3)},{y,0,Sqrt[(n-x^3)/2]}];Print[n," ",r];Continue,{n,0,80}]

A115159 Numbers that are not the sum of a triangular number, a square and a fourth power.

Original entry on oeis.org

34, 63, 89, 99, 139, 164, 174, 193, 204, 245, 314, 399, 424, 454, 464, 489, 504, 524, 549, 714, 1049, 1149, 1174, 1439, 1504, 1539, 1639, 1799, 1814, 1919, 2164, 2239, 2313, 2374, 2414, 2439, 2764, 2789, 3079, 3319, 3414, 3669, 3774, 3814, 4019, 4114

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

There are 718 such numbers up to 2*10^9, the last one in this range being 99570649.

It is known that each natural number can be written as the sum of two squares and a triangular number. I believe that the sequence only has 718 terms as found by _Giovanni Resta and listed in the b-file. - Zhi-Wei Sun, Apr 15 2020

Zhi-Wei Sun, Table of n, a(n) for n = 1..718
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Cf. A022552, A014158, A115160, A115161, A115162, A115163, A262959, A334086, A334113.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[Do[If[TQ[n-x^4-y^2],Goto[aa]],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]}];tab=Append[tab,n];Label[aa],{n,0,4114}];Print[tab] (From Zhi-Wei Sun)

Definition corrected by Giovanni Resta, Aug 17 2011

A115162 Positive numbers that are not the sum of a triangular number, a square and a cube, all of them greater than or equal to 1.

Original entry on oeis.org

1, 2, 4, 7, 9, 14, 21, 28, 35, 70, 126, 152, 161, 481

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

Probably finite. No other terms up to 10^9.

Cf. A022552, A014158, A115159, A115160, A115161, A115163.

A115161 Numbers that are not the sum of a triangular number, a nonnegative cube and a fourth power.

Original entry on oeis.org

13, 35, 40, 41, 50, 51, 59, 76, 77, 112, 124, 139, 149, 150, 157, 165, 168, 175, 176, 178, 182, 183, 189, 193, 197, 205, 208, 215, 224, 229, 241, 243, 249, 273, 288, 305, 306, 314, 329, 332, 366, 373, 383, 397, 412, 413, 418, 420, 429, 438, 453, 455, 457, 461

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

There are 15682461 such numbers up to 10^9, the last one in this range being 999999923.

Cf. A022552, A014158, A115159, A115160, A115162, A115163.

A115163 Numbers that are not the sum of two triangular numbers and a nonnegative cube.

Original entry on oeis.org

41, 104, 188, 923

Offset: 1

Giovanni Resta, Jan 15 2006

nonn

Probably finite. No other terms up to 10^9.

Cf. A022552, A014158, A115159, A115160, A115161, A115162.

A275169 Positive integers not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

Original entry on oeis.org

15, 21, 47, 53, 79, 85, 92, 111, 117, 120, 181, 183, 245, 309, 311, 335, 372, 373, 398, 405, 421, 437, 447, 501, 565, 573, 629, 636, 645, 655, 693, 757, 791, 807, 820, 821, 853, 869, 885, 888, 949, 967, 1013, 1045, 1077, 1141, 1205, 1223, 1269, 1271, 1303, 1461, 1555, 1591, 1613, 1653, 2087, 2101, 2255, 2421

Offset: 1

Zhi-Wei Sun, Jul 18 2016

nonn

Conjecture: The sequence has totally 174 terms as listed in the b-file the largest of which is 375565.
This implies the conjecture in A275150. We note that the sequence contains no term greater than 375565 and not exceeding 10^6.
See also A275168 for a similar conjecture.

			a(1) = 15 since 15 is the least positive integer not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

Zhi-Wei Sun, Table of n, a(n) for n = 1..174

Cf. A000290, A000578, A022551, A022552, A262813, A270488, A274274, A275083, A275150, A275168.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0;Do[Do[If[SQ[m-x^3-2*y^2],Goto[aa]],{x,0,m^(1/3)},{y,0,Sqrt[(m-x^3)/2]}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,1,2421}]

A297970 Numbers that are not the sum of 3 squares and a nonnegative 7th power.

Original entry on oeis.org

112, 240, 368, 496, 624, 752, 880, 1008, 1136, 1264, 1392, 1520, 1648, 1776, 1904, 2032, 2160

Offset: 1

XU Pingya, Jan 10 2018

The last term in this sequence is 2160. The reasons are as follows (let b, c, d, i, j, k, m, r, s, t, w, x, y and z be nonnegative integers).
For the Diophantine equation x^2 + y^2 + z^2 + w^7 = m:
(1) If m is not of the form 4^c * (8b + 7), then it follows from Legendre's three-square theorem that the equation has a solution with w = 0.
(2) 8b + 7 - 1^7 = 8b + 6. Then m = 8b + 7, the equation has a solution with w = 1.
(3) 4 * (8b + 7) - 1^7 = (8 * (4b + 3) + 3) = 8d + 3. Then m = 4 * (8b + 7), the equation has a solution with w = 1.
(4) For b >= 17, 16 * (8b + 7) - 3^7 = 8 * (16 * (b - 17) + 12) + 5 = 8i + 5. Then m = 16 * (8b + 7) and b >= 17, the equation has a solution with w = 3.
(5) 4^3 * (8b + 7) - 2^7 = 4^3 * (8b + 5). Then m = 4^3 * (8b + 7), the equation has a solution with w = 2. And 4^3 * (8b + 7) - 3^7 = 8 * (4^3 * (b - 4) + 38) + 5 = 8j + 5. Then m = 4^3 * (8b + 7) and b >= 4, the equation has a solution with w = 3.
(6) 4^4 * (8b + 7) - 2^7 = 4^3 * (8 * (4b + 3) + 3) = 4^3 * (8k + 3). 4^4 * (8b + 7) - 3^7 = 8 * (256b - 217) + 3 = 8r + 3. Then m = 4^4 * (8b + 7), the equation has a solution with w = 2 and when b > 0, the equation has a solution with w = 3.
(7) When c >= 5, 4^c * (8b + 7) - 2^7 = 4^3 * (8 * (b * 4^(c - 3) + 14 * 4^(c - 5) + 5) = 4^3 * (8s + 5). 4^c * (8b + 7) - 3^7 = 8 * (b * 4^(c - 3) + 14 * 4^(c - 3) - 273) + 3 = 8t + 3. Then n = 4^c * (8b + 7), the equation has solutions with w = 2 and 3.
In short, except for the 17 numbers in the sequence, every nonnegative integer can be represented as the sum of 3 squares and a nonnegative 7th power.

Wikipedia, Legendre's three-square theorem

Finite subsequence of A004215 and A296185.

Cf. A004771, A022552, A022557, A022561, A022566, A111151.

t1={};
Do[Do[If[x^2+y^2+z^2+w^7==n, AppendTo[t1,n]&&Break[]], {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,y,(n-x^2-y^2)^(1/2)}, {w,0,(n-x^2-y^2-z^2)^(1/7)}], {n,0,3000}];
t2={};
Do[If[FreeQ[t1,k]==True, AppendTo[t2,k]], {k,0,3000}];
t2

a(n) = 128n - 16 = 16 * A004771(n - 1), 1 <= n <= 17.

A275083 Positive integers congruent to 0 or 1 modulo 4 that cannot be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers.

Original entry on oeis.org

120, 312, 813, 2136, 2680, 3224, 4404, 5340, 6420, 10060, 11320, 11824, 14008, 15856, 26544, 28804, 34392, 47984

Offset: 1

Zhi-Wei Sun, Jul 15 2016

nonn

Conjecture: (i) The sequence has totally 18 terms as listed.
(ii) For each r = 2,3 there are infinitely many positive integers n == r (mod 4) not in the form x^3 + y^2 + z^2 with x,y,z nonnegative integers.
Our computation indicates that the sequence has no other terms below 10^6.
Let d be 2 or 6. Clearly, n-d is congruent to 0 or 1 modulo 4 if n is congruent to 2 or 3 modulo 4. So part (i) of the conjecture essentially implies that for each n = 0,1,2,... either n or n-d can be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers.

			a(1) = 120 since all those positive integers congruent to 0 or 1 modulo 4 and smaller than 120 can be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers but 120 (divisible by 4) cannot be written in this way.

Cf. A000290, A000578, A022551, A022552, A262813, A262857, A272979, A274274.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0;Do[If[Mod[m,4]>1,Goto[aa]];Do[If[SQ[m-x^3-y^2],Goto[aa]],{x,0,m^(1/3)},{y,0,Sqrt[(m-x^3)/2]}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,1,50000}]

cp's OEIS Frontend

A022551 Numbers that are the sum of 2 squares and a nonnegative cube.

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A115160 Numbers that are not the sum of two triangular numbers and a fourth power.

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A274274 Number of ordered ways to write n as x^3 + y^2 + z^2, where x,y,z are nonnegative integers with y <= z.

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A115159 Numbers that are not the sum of a triangular number, a square and a fourth power.

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A115162 Positive numbers that are not the sum of a triangular number, a square and a cube, all of them greater than or equal to 1.

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A115161 Numbers that are not the sum of a triangular number, a nonnegative cube and a fourth power.

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A115163 Numbers that are not the sum of two triangular numbers and a nonnegative cube.

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A275169 Positive integers not in the form x^3 + 2*y^2 + z^2 with x,y,z nonnegative integers.

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A297970 Numbers that are not the sum of 3 squares and a nonnegative 7th power.

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A275083 Positive integers congruent to 0 or 1 modulo 4 that cannot be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers.

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