A271237 -id:A271237 - cp's OEIS frontend

A267826 Numbers not of the form w^3 + 2x^3 + 3y^3 + 4*z^3, where w, x, y and z are nonnegative integers.

18, 22, 39, 60, 63, 74, 76, 77, 100, 103, 106, 107, 117, 126, 178, 180, 201, 215, 228, 230, 245, 271, 289, 291, 295, 315, 341, 356, 357, 393, 413, 419, 420, 480, 481, 523, 559, 606, 616, 671, 673, 705, 854, 855, 963, 980, 981, 998, 1103, 1121, 1130, 1298, 1484, 1510, 1643, 1729, 1849, 1916, 1934, 1946

Offset: 1

Zhi-Wei Sun, Apr 07 2016

nonn

Conjecture: The sequence has exactly 122 terms the last of   which is a(122) = 41405.
We have verified that there are no terms between 41406 and 2*10^5.
The conjecture implies that {P(v)+w^3+2*x^3+3*y^3+4*z^3: w,x,y,z = 0,1,2,...} = {0,1,2,...} whenever P(v) is among the polynomials a*v^3 (a = 1,5,6,7,9,10,12,15,18), b*v^4 (b = 1,2,3,5,6,12,18), c*v^5 (c = 1,2,5,12) and d*v^k (d = 5,12; k = 6,7). Moreover, it also implies that {8*t+w^3+2*x^3+3*y^3+4*z^3: t = 0,1; w,x,y,z = 0,1,2,...} = {0,1,2,...}. If a,b,c,d and m are positive integers with {m*t+a*w^3+b*x^3+c*y^3+d*z^3: t = 0,1; w,x,y,z = 0,1,2,...} = {0,1,2,...}, then we must have m = 8 and {a,b,c,d} = {1,2,3,4}.

			a(1) = 18 since it is the first nonnegative integer not in the set {w^3 + 2*x^3 + 3*y^3 + 4*z^3: w,x,y,z = 0,1,2,...}.

Zhi-Wei Sun, Table of n, a(n) for n = 1..122
Zhi-Wei Sun, Universal sums u^3+a*v^3+b*x^3+c*y^3+d*z^3 with u, v, x, y, z nonnegative integers, a message to Number Theory Mailing List, April 3, 2016.

Cf. A000578, A002804, A022566, A267861, A271099, A271169, A271237.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
n=0;Do[Do[If[CQ[m-4*z^3-3y^3-2x^3],Goto[aa]],{z,0,(m/4)^(1/3)},{y,0,((m-4z^3)/3)^(1/3)},{x,0,((m-4z^3-3y^3)/2)^(1/3)}];n=n+1;Print[n," ",m];Label[aa];Continue,{m,0,1946}]

A267861 Number of ways to write n as 2t + u^4 + v^4 + 2w^4 + 3x^4 + 4y^4 + 6*z^4, where t is 0 or 1, and u, v, w, x, y, z are nonnegative integers with u <= v and v > 0.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 5, 5, 5, 4, 3, 4, 3, 3, 4, 3, 4, 5, 5, 5, 6, 5, 5, 5, 4, 3, 3, 3, 2, 4, 2, 4, 4, 5, 5, 6, 5, 5, 6, 4, 4, 3, 3, 2, 4, 2, 4, 4, 4, 5, 6, 5, 6, 6, 4, 4, 4, 3, 2, 4, 2, 4, 5, 6, 5, 8

Offset: 1

Zhi-Wei Sun, Apr 07 2016

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 1, 2, 111, 127, 143, 158, 221, 223, 240, 460, 463, 480, 545, 560, 561, 1455, 1695, 1776, 2175. Moreover, any integer n > 10^4 not among 10543, 17935, 37583, 40383, 78543 can be written as u^4 + v^4 + 2*w^4 + 3*x^4 + 4*y^4 + 6*z^4 with u,v,w,x,y,z nonnegative integers.

If a(1),...,a(7) are positive integers with a(1) <= a(2) <= ... <= a(7) and a(1)+...+a(7) = g(4) = 19 such that {a(1)*x(1)^4+...+a(7)*x(7)^4: x(1),...,x(7) = 0,1,2,...} = {0,1,2,...}, then the tuple (a(1),...,a(7)) must be (1,1,2,2,3,4,6) or (1,1,2,2,3,3,7). Similarly, if a(1),...,a(8) are positive integers with a(1) <= a(2) <= ... <= a(8) and a(1)+...+a(8) = g(5) = 37 such that {a(1)*x(1)^5+...+a(8)*x(8)^5: x(1),...,x(8) = 0,1,2,...} = {0,1,2,...}, then (a(1),...,a(8)) must be (1,1,2,3,4,6,8,12) or (1,1,2,3,4,5,7,14).

			a(111) = 1 since 111 = 2*1 + 2^4 + 3^4 + 2*1^4 + 3*0^4 + 4*1^4 + 6*1^4.
a(240) = 1 since 240 = 2*0 + 2^4 + 2^4 + 2*0^4 + 3*2^4 + 4*2^4 + 6*2^4.
a(1776) = 1 since 1776 = 2*0 + 4^4 + 5^4 + 2*3^4 + 3*3^4 + 4*1^4 + 6*3^4.
a(2175) = 1 since 2175 = 2*1 + 0^4 + 4^4 + 2*2^4 + 3*5^4 + 4*1^4 + 6*1^4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Upgrade Waring's Problem, a message to Number Theory Mailing List, April 2, 2016.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000583, A000584, A002804, A267826, A271099, A271169, A271237.

QQ[n_]:=QQ[n]=n>0&&IntegerQ[n^(1/4)]
Do[r=0;Do[If[QQ[n-2t-6*z^4-4y^4-3x^4-2w^4-u^4],r=r+1],{t,0,Min[1,n/2]},{z,0,((n-2t^8)/6)^(1/4)},{y,0,((n-2t-6z^4)/4)^(1/4)},{x,0,((n-2t-6z^4-4y^4)/3)^(1/4)},
{w,0,((n-2t-6z^4-4y^4-3x^4)/2)^(1/4)},{u,0,((n-2t-6z^4-4y^4-3x^4-2w^4)/2)^(1/4)}];Print[n," ",r];Continue,{n,1,70}]

A271076 Number of ordered ways to write n as u^5 + v^4 + x^3 + 2y^3 + 3z^3, where u, v , x, y and z are nonnegative integers with v > 0.

Original entry on oeis.org

1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 4, 4, 4, 4, 3, 1, 1, 3, 4, 4, 5, 4, 2, 2, 2, 5, 4, 3, 5, 2, 1, 1, 2, 5, 4, 6, 5, 2, 3, 2, 4, 5, 4, 3, 3, 3, 2, 2, 4, 5, 4, 5, 5, 1, 2, 3, 3, 5, 2, 5, 5, 3, 3, 3, 3, 3, 4, 4, 1, 1, 2, 3, 5

Offset: 1

Zhi-Wei Sun, Apr 07 2016

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 8, 9, 10, 11, 13, 14, 15, 16, 23, 24, 38, 39, 61, 76, 77, 104, 118, 188, 214, 229.
(ii) We have {P(u,v,x,y,z): u,v,x,y,z = 0,1,2,...} = {0,1,2,...} whenever P(u,v,x,y,z) is among the following polynomials: u^5+v^4+x^3+2*y^3+5*z^3, u^5+2*v^4+x^3+2*y^3+3*z^3, u^5+3*v^4+x^3+2*y^3+3*z^3, 2*u^5+v^4+x^3+y^3+4*z^3, 2*u^5+v^4+x^3+2*y^3+4*z^3, 3*u^5+v^4+x^3+2*y^3+4*z^3, 5*u^5+v^4+x^3+2*y^3+4*z^3, u^4+2*v^4+x^3+y^3+4*z^3, u^4+2*v^4+x^3+2*y^3+3*z^3, u^4+2*v^4+x^3+2*y^3+4*z^3, u^4+2*v^4+x^3+2*y^3+6*z^3, u^4+2*v^4+x^3+3*y^3+4*z^3, u^4+2*v^4+x^3+4*y^3+5*z^3, u^4+2*v^4+x^3+4*y^3+6*z^3, u^4+2*v^4+x^3+4*y^3+10*z^3, u^4+3*v^4+x^3+2*y^3+3*z^3, u^4+3*v^4+x^3+2*y^3+4*z^3, u^4+3*v^4+x^3+2*y^3+6*z^3, u^4+4*v^4+x^3+y^3+2*z^3, u^4+4*v^4+x^3+2*y^3+3*z^3, u^4+4*v^4+x^3+2*y^3+4*z^3, u^4+5*v^4+x^3+2*y^3+4*z^3, u^4+6*v^4+x^3+2*y^3+3*z^3, u^4+7*v^4+x^3+2*y^3+3*z^3, u^4+9*v^4+x^3+2*y^3+4*z^3, 2*u^4+4*v^4+x^3+2*y^3+3*z^3,2*u^4+6*v^4+x^3+2*y^3+4*z^3, 3*u^4+6*v^4+x^3+2*y^3+4*z^3.
(iii) We have {P(u,v,x,y,z): u,v,x,y,z = 0,1,2,...} = {0,1,2,...} whenever P(u,v,x,y,z) is among the following polynomials: u^5+v^3+x^3+2*y^3+4*z^3, u^5+v^3+2*x^3+3*y^3+c*z^3 (c = 6,9), a*u^5+v^3+2*x^3+4*y^3+5*z^3 (a = 1,2,6), b*u^5+v^3+2*x^3+4*y^3+6*z^3 (b = 2,3), u^5+v^3+2*x^3+4*y^3+d*z^3 (d = 9,13).
The listed polynomials in part (ii), together with u^5+v^4+x^3+2*y^3+3*z^3, should essentially exhaust all those polynomials P(u,v,x,y,z) = s*u^k+t*v^j+a*x^3+b*y^3+c*z^3 with s,t,a,b,c positive integers and k >= j > 3, such that {P(u,v,x,y,z): u,v,x,y,z = 0,1,2,...} = {0,1,2,...}.
There are also finitely many (but quite a lot) polynomials P(u,v,x,y,z) of the form m*u^4+a*v^3+b*x^3+c*y^3+d*z^3 with a,b,c,d and m positive integers such that {P(u,v,x,y,z): u,v,x,y,z = 0,1,2,...} = {0,1,2,...}.
See also A267826, A271099 and A271237 for related comments.
Conjectures (i), (ii) and (iii) verified for n up to 10^11 for all polynomials. - Mauro Fiorentini, Sep 20 2023

			a(16) = 1 since 16 = 0^5 + 2^4 +0^3 + 2*0^3 + 3*0^3.
a(104) = 1 since 104 = 0^5 + 2^4 + 4^3 + 2*0^3 + 3*2^3.
a(188) = 1 since 188 = 2^5 + 1^4 + 3^3 + 2*4^3 + 3*0^3.
a(229) = 1 since 229 = 1^5 + 3^4 + 4^3 + 2*1^3 + 3*3^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000578, A000583, A000584, A266968, A267826, A271099, A271237.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[r=0;Do[If[CQ[n-u^5-v^4-3z^3-2y^3],r=r+1],{u,0,(n-1)^(1/5)},{v,1,(n-u^5)^(1/4)},{z,0,((n-u^5-v^4)/3)^(1/3)},{y,0,((n-u^5-v^4-3z^3)/2)^(1/3)}];Print[n," ",r];Continue,{n,1,80}]

A352503 Number of ways to write n as w^3 + 2x^3 + 4y^3 + 5*z^3 + t^6, where w is a positive integer, and x,y,z,t are nonnegative integers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 3, 2, 1, 2, 4, 4, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 5, 3, 2, 1, 2, 2, 2, 3, 2, 2, 1, 2, 3, 4, 4, 1, 4, 5, 3, 6, 4, 5, 4, 5, 5, 3, 5, 3, 5, 1, 1, 1, 3, 6, 2, 3, 2, 4, 4, 3, 3, 2, 4, 2, 2, 3, 1, 3, 4, 5, 2, 5, 4

Offset: 1

Zhi-Wei Sun, Mar 28 2022

nonn

Conjecture: a(n) > 0 for all n > 0.
This has been verified for n = 1..10^6.
It seems that a(n) = 1 only for n = 1..5, 16, 19, 20, 21, 23, 24, 25, 26, 31, 37, 51, 58, 63, 77, 78, 79, 94, 108, 207, 208, 218, 316, 487, 490, 559.

			a(20) = 1 with 20 = 2^3 + 2*1^3 + 4*1^3 + 5*1^3 + 1^6.
a(79) = 1 with 79 = 2^3 + 2*1^3 + 4*0^3 + 5*1^3 + 2^6.
a(316) = 1 with 316 = 1^3 + 2*3^3 + 4*4^3 + 5*1^3 + 0^6.
a(487) = 1 with 487 = 5^3 + 2*!^3 + 4*4^3 + 5*2^3 + 2^6.
a(490) = 1 with 490 = 2^3 + 2*3^3 + 4*3^3 + 5*4^3 + 0^6.
a(559) = 1 with 559 = 8^3 + 2*1^3 + 4*1^3 + 5*2^3 + 1^6.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120. (See Conjecture 3.4(i).)

Cf. A000578, A001014, A271099, A271237.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={};Do[r=0;Do[If[CQ[n-t^6-2x^3-4y^3-5z^3],r=r+1],{t,0,(n-1)^(1/6)},{x,0,((n-1-t^6)/2)^(1/3)},{y,0,((n-1-t^6-2x^3)/4)^(1/3)},{z,0,((n-1-t^6-2x^3-4y^3)/5)^(1/3)}];tab=Append[tab,r],{n,1,100}];Print[tab]

cp's OEIS Frontend

A267826 Numbers not of the form w^3 + 2*x^3 + 3*y^3 + 4*z^3, where w, x, y and z are nonnegative integers.

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A267861 Number of ways to write n as 2*t + u^4 + v^4 + 2*w^4 + 3*x^4 + 4*y^4 + 6*z^4, where t is 0 or 1, and u, v, w, x, y, z are nonnegative integers with u <= v and v > 0.

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A271076 Number of ordered ways to write n as u^5 + v^4 + x^3 + 2*y^3 + 3*z^3, where u, v , x, y and z are nonnegative integers with v > 0.

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A352503 Number of ways to write n as w^3 + 2*x^3 + 4*y^3 + 5*z^3 + t^6, where w is a positive integer, and x,y,z,t are nonnegative integers.

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A267826 Numbers not of the form w^3 + 2x^3 + 3y^3 + 4*z^3, where w, x, y and z are nonnegative integers.

A267861 Number of ways to write n as 2t + u^4 + v^4 + 2w^4 + 3x^4 + 4y^4 + 6*z^4, where t is 0 or 1, and u, v, w, x, y, z are nonnegative integers with u <= v and v > 0.

A271076 Number of ordered ways to write n as u^5 + v^4 + x^3 + 2y^3 + 3z^3, where u, v , x, y and z are nonnegative integers with v > 0.

A352503 Number of ways to write n as w^3 + 2x^3 + 4y^3 + 5*z^3 + t^6, where w is a positive integer, and x,y,z,t are nonnegative integers.