A303338 -id:A303338 - cp's OEIS frontend

A303233 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

0, 1, 2, 3, 4, 5, 4, 6, 7, 7, 7, 9, 7, 8, 9, 9, 8, 12, 11, 11, 11, 11, 11, 14, 11, 13, 12, 11, 10, 14, 11, 12, 17, 15, 12, 16, 14, 15, 17, 19, 15, 16, 13, 15, 17, 17, 16, 20, 16, 14, 17, 17, 14, 22, 17, 14, 14, 17, 15, 19

Offset: 1

Zhi-Wei Sun, Apr 20 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two triangular numbers and two powers of 2.
a(n) > 0 for all n = 2..10^9. See A303234 for numbers of the form x*(x+1)/2 + 2^y with x and y nonnegative integers. See also A303363 for a stronger conjecture.
In contrast, Crocker proved in 2008 that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of 2.

			a(2) = 1 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^0.
a(3) = 2 with 3 = 0*(0+1)/2 + 1*(1+1)/2 + 2^0 + 2^0 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^1.
a(4) = 3 with 4 = 1*(1+1)/2 + 1*(1+1)/2 + 2^0 + 2^0 = 0*(0+1)/2 + 1*(1+1)/2 + 2^0 + 2^1 = 0*(0+1)/2 + 0*(0+1)/2 + 2^1 + 2^1.

R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000217, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303234, A303338, A303363.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-2^k-2^j)+1],Do[If[SQ[8(n-2^k-2^j-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-2^k-2^j)+1]-1)/2}]],{k,0,Log[2,n]-1},{j,k,Log[2,n-2^k]}];tab=Append[tab,r],{n,1,60}];Print[tab]

A303363 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 2, 4, 6, 3, 5, 6, 4, 6, 7, 4, 4, 9, 6, 6, 8, 4, 9, 9, 5, 7, 7, 5, 7, 9, 4, 8, 13, 7, 6, 11, 7, 10, 13, 8, 9, 10, 7, 9, 11, 7, 9, 15, 8, 8, 14, 6, 9, 16, 6, 8, 11, 11, 10, 12, 8, 7, 15, 10, 8, 11, 9, 14, 15, 9

Offset: 1

Zhi-Wei Sun, Apr 22 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This is stronger than the author's conjecture in A303233. I have verified a(n) > 0 for all n = 2..10^9.
In contrast, Corcker proved in 2008 that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of 2.

			a(2) = 1 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^0.
a(3) = 2 with 3 = 0*(0+1)/2 + 1*(1+1)/2 + 2^0 + 2^0 = 0*(0+1)/2 + 0*(0+1)/2 + 2^0 + 2^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000079, A000217, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303389.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-4^j-2^k)+1],Do[If[SQ[8(n-4^j-2^k-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-4^j-2^k)+1]-1)/2}]],{j,0,Log[4,n/2]},{k,2j,Log[2,n-4^j]}];tab=Append[tab,r],{n,1,70}];Print[tab]

A303389 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c + 5^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 2, 2, 2, 4, 3, 2, 2, 3, 3, 3, 2, 2, 2, 4, 3, 2, 1, 5, 4, 3, 2, 5, 5, 5, 5, 3, 3, 5, 5, 4, 4, 4, 5, 5, 2, 5, 3, 5, 4, 7, 2, 4, 6, 6, 5, 4, 4, 5, 8, 4, 4, 4, 7, 6, 4, 3, 4, 8, 4, 7, 3, 3, 6, 8, 2, 5, 6, 5, 4, 6, 4, 3

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integers n > 1 can be written as the sum of two triangular numbers and two powers of 5.
This has been verified for all n = 2..10^10.
See A303393 for the numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.
See also A303401, A303432 and A303540 for similar conjectures.

			a(4) = 1 with 4 = 1*(1+1)/2 + 1*(1+1)/2 + 5^0 + 5^0.
a(5) = 1 with 5 = 0*(0+1)/2 + 2*(2+1)/2 + 5^0 + 5^0.
a(7) = 1 with 7 = 0*(0+1)/2 + 1*(1+1)/2 + 5^0 + 5^1.
a(25) = 1 with 25 = 0*(0+1)/2 + 5*(5+1)/2 + 5^1 + 5^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000217, A000351, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303363, A303393, A303401, A303432, A303540.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-5^j-5^k)+1],Do[If[SQ[8(n-5^j-5^k-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-5^j-5^k)+1]-1)/2}]],{j,0,Log[5,n/2]},{k,j,Log[5,n-5^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A303393 Numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 11, 15, 16, 20, 22, 25, 26, 28, 29, 31, 33, 35, 37, 40, 41, 46, 50, 53, 56, 60, 61, 67, 70, 71, 79, 80, 83, 91, 92, 96, 103, 106, 110, 116, 121, 125, 126, 128, 130, 131, 135, 137, 140, 141, 145, 146, 153, 154, 158, 161, 170, 172, 176

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

The author's conjecture in A303389 has the following equivalent version: Each integer n > 1 can be expressed as the sum of two terms of the current sequence.

This has been verified for all n = 2..2*10^8.

			a(1) = 1 with 1 = 0*(0+1)/2 + 5^0.
a(2) = 2 with 2 = 1*(1+1)/2 + 5^0.
a(3) = 4 with 4 = 2*(2+1)/2 + 5^0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000217, A000351, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303363, A303389.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={};Do[Do[If[TQ[m-5^k],tab=Append[tab,m];Goto[aa]],{k,0,Log[5,m]}];Label[aa],{m,1,176}];Print[tab]

A303399 Number of ordered pairs (a, b) with 0 <= a <= b such that n - 5^a - 5^b can be written as the sum of two triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 1, 4, 3, 3, 2, 5, 4, 4, 4, 3, 3, 4, 4, 3, 4, 4, 4, 3, 2, 4, 3, 3, 3, 5, 2, 4, 5, 4, 4, 4, 4, 3, 5, 3, 4, 4, 4, 4, 4, 3, 3, 5, 4, 5, 3, 3, 5, 5, 2, 4, 6, 3, 3, 4, 4, 3

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This is equivalent to the author's conjecture in A303389. It has been verified that a(n) > 0 for all n = 2..6*10^9.
Note that a nonnegative integer m is the sum of two triangular numbers if and only if 4*m + 1 can be written as the sum of two squares.

			a(6) = 2 with 6 - 5^0 - 5^0 = 1*(1+1)/2 + 2*(2+1)/2 and 6 - 5^0 - 5^1 = 0*(0+1)/2 + 0*(0+1)/2.
a(7) = 1 with 7 - 5^0 - 5^1 = 0*(0+1)/2 + 1*(1+1)/2.
a(25) = 1 with 25 - 5^1 - 5^1 = 0*(0+1)/2 + 5*(5+1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000217, A000351, A271518, A273812, A281976, A299924, A300219, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303363, A303389, A303393.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-5^j-5^k)+1],r=r+1],{j,0,Log[5,n/2]},{k,j,Log[5,n-5^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A303401 Number of ways to write n as a(3a-1)/2 + b(3b-1)/2 + 3^c + 3^d with a,b,c,d nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two pentagonal numbers and two powers of 3.

a(n) > 0 for all n = 2..7*10^6. See A303434 for the numbers of the form x*(3*x-1)/2 + 3^y with x and y nonnegative integers. See also A303389 and A303432 for similar conjectures.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000244, A000326, A303233, A303338, A303363, A303389, A303393, A303399, A303428, A303432, A303434.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PenQ[n_]:=PenQ[n]=SQ[24n+1]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0);
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[12(n-3^j-3^k)+1],Do[If[PenQ[n-3^j-3^k-x(3x-1)/2],r=r+1],{x,0,(Sqrt[12(n-3^j-3^k)+1]+1)/6}]],{j,0,Log[3,n/2]},{k,j,Log[3,n-3^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

a(78) = 1 with 78 = 3*(3*3-1)/2 + 3*(3*3-1)/2 + 3^3 + 3^3.
a(285) = 1 with 285 = 3*(3*1-1)/2 + 11*(3*11-1)/2 + 3^3 + 3^4.
a(711) = 1 with 711 = 9*(3*9-1)/2 + 20*(3*20-1)/2 + 3^0 + 3^1.
a(775) = 1 with 775 = 7*(3*7-1)/2 + 21*(3*21-1)/2 + 3^3 + 3^3.
a(3200) = 1 with 12*(3*12-1)/2 + 44*(3*44-1)/2 + 3^3 + 3^4.
a(13372) = 1 with 13372 = 17*(3*17-1)/2 + 65*(3*65-1)/2 + 3^4 + 3^8.
a(16545) = 1 with 16545 = 0*(3*0-1)/2 + 98*(3*98-1)/2 + 3^0 + 3^7.

A303432 Number of ways to write n as a(2a-1) + b(2b-1) + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 2, 3, 4, 5, 4, 4, 2, 3, 3, 4, 5, 7, 5, 5, 4, 4, 4, 7, 5, 4, 3, 2, 2, 4, 5, 7, 8, 7, 5, 7, 5, 7, 7, 7, 4, 4, 2, 3, 5, 7, 6, 9, 7, 6, 5, 6, 5, 7, 7, 3, 3, 3, 3, 5, 7, 7, 8, 7, 6, 8, 5, 8, 8, 8, 5, 7, 4, 6, 7, 9, 8, 9, 7, 8

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture 1: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two hexagonal numbers and two powers of 2.
Conjecture 2: Any integer n > 1 can be written as a*(2*a+1) + b*(2*b+1) + 2^c + 2^d with a,b,c,d nonnegative integers.
Conjecture 3: Each integer n > 1 can be written as a*(2*a-1) + b*(2*b+1) + 2^c + 2^d with a,b,c,d nonnegative integers.
All the three conjectures hold for n = 2..2*10^6. Note that either of them is stronger than the conjecture in A303233.
See also A303363, A303389 and  A303401 for similar conjectures.

			a(2) = 1 with 2 = 0*(2*0-1) + 0*(2*0-1) + 2^0 + 2^0.
a(7) = 2 with 7 = 1*(2*1-1) + 1*(2*1-1) + 2^0 + 2^2 = 0*(2*0-1) + 1*(2*1-1) + 2^1 + 2^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000079, A000384, A014105, A303233, A303338, A303363, A303389, A303393, A303399, A303401.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
HexQ[n_]:=HexQ[n]=SQ[8n+1]&&(n==0||Mod[Sqrt[8n+1]+1,4]==0);
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[4(n-2^j-2^k)+1],Do[If[HexQ[n-2^j-2^k-x(2x-1)],r=r+1],{x,0,(Sqrt[4(n-2^j-2^k)+1]+1)/4}]],{j,0,Log[2,n/2]},{k,j,Log[2,n-2^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A303540 Number of ways to write n as a^2 + b^2 + binomial(2c,c) + binomial(2d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

Original entry on oeis.org

0, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 6, 4, 2, 2, 4, 4, 2, 2, 5, 5, 5, 4, 4, 4, 4, 5, 6, 5, 5, 4, 5, 4, 4, 3, 4, 5, 5, 6, 5, 5, 5, 4, 7, 3, 4, 5, 6, 4, 2, 4, 6, 7, 4, 4, 5, 7, 6, 2, 5, 4, 6, 3, 2, 5, 5, 5, 4, 4, 3, 7, 9, 6, 5, 6, 11, 7, 3, 4, 8

Offset: 1

Zhi-Wei Sun, Apr 25 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two squares and two central binomial coefficients.
It has been verified that a(n) > 0 for all n = 2..10^10.
See also A303539 and A303541 for related information.
Jiao-Min Lin (a student at Nanjing University) has verified a(n) > 0 for all 1 < n <= 10^11. - Zhi-Wei Sun, Jul 30 2022

			a(2) = 1 since 2 = 0^2 + 0^2 + binomial(2*0,0) + binomial(2*0,0).
a(10) = 2 with 10 = 2^2 + 2^2 + binomial(2*0,0) + binomial(2*0,0) = 1^2 + 1^2 + binomial(2*1,1) + binomial(2*2,2).
a(2435) = 1 with 2435 = 32^2 + 33^2 + binomial(2*4,4) + binomial(2*5,5).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000290, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303541, A303543, A303601, A303639.

N:= 100: # for a(1)..a(N)
A:= Vector(N):
for b from 0 to floor(sqrt(N)) do
  for a from 0 to min(b, floor(sqrt(N-b^2))) do
    t:= a^2+b^2;
    for d from 0 do
      s:= t + binomial(2*d,d);
      if s > N then break fi;
      for c from 0 to d do
        u:= s + binomial(2*c,c);
        if u > N then break fi;
        A[u]:= A[u]+1;
od od od od:
convert(A,list); # Robert Israel, May 30 2018

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
c[n_]:=c[n]=Binomial[2n,n];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;k=0;Label[bb];If[c[k]>n,Goto[aa]];Do[If[QQ[n-c[k]-c[j]],Do[If[SQ[n-c[k]-c[j]-x^2],r=r+1],{x,0,Sqrt[(n-c[k]-c[j])/2]}]],{j,0,k}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,80}];Print[tab]

A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 1, 3, 3, 5, 2, 3, 3, 2, 2, 5, 4, 5, 2, 3, 5, 2, 3, 5, 4, 7, 2, 4, 5, 3, 4, 6, 4, 7, 3, 6, 6, 4, 4, 5, 5, 9, 5, 6, 6, 2, 5, 5, 7, 8, 4, 5, 4, 4, 4, 6, 6, 8, 3, 6, 6, 3, 4, 6, 7, 8, 5, 8, 6, 5, 4, 6, 7, 8, 6, 6, 6, 2

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1. Moreover, any integer n > 1 can be written as x*(3*x+2) + y*(3*y+2) + 3^z + 3^w, where x is an integer and y,z,w are nonnegative integers.
a(n) > 0 for all n = 2..3*10^8. Those x*(3*x-2) with x integral are called generalized octagonal numbers (A001082). 76683391 is the least integer n > 1 not representable as the sum of two generalized octagonal numbers and two powers of 2.
See also A303389, A303401 and A303432 for similar conjectures.

			a(2) = 1 with 2 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^0.
a(3) = 1 with 3 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^0.
a(4) = 2 with 4 = 1*(3*1-2) + 1*(3*1-2) + 3^0 + 3^0 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^1.
a(5) = 1 with 5 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^1.
a(9) = 1 with 9 = (-1)*(3*(-1)-2) + 0*(3*0-2) + 3^0 + 3^1.
a(4360) = 4 with 4360 = (-35)*(3*(-35)-2) + (-13)*(3*(-13)-2) + 3^0 + 3^4 = (-37)*(3*(-37)-2) + (-7)*(3*(-7)-2) + 3^2 + 3^2 = (-27)*(3*(-27)-2) + (-23)*(3*(-23)-2) + 3^5 + 3^5 = (-25)*(3*(-25)-2) + (-1)*(3*(-1)-2) + 3^5 + 3^7.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000244, A000567, A001082, A045944, A280472, A303233, A303338, A303363, A303389, A303393, A303399, A303401, A303432.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[3(n-3^j-3^k)+2],Do[If[SQ[3(n-3^j-3^k-x(3x-2))+1],r=r+1],{x,-Floor[(Sqrt[3(n-3^j-3^k)/2+1]-1)/3],(Sqrt[3(n-3^j-3^k)/2+1]+1)/3}]],
{j,0,Log[3,n/2]},{k,j,Log[3,n-3^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A303434 Numbers of the form x(3x-1)/2 + 3^y with x and y nonnegative integers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 13, 14, 15, 21, 23, 25, 27, 28, 31, 32, 36, 38, 39, 44, 49, 52, 54, 60, 62, 71, 73, 78, 79, 81, 82, 86, 93, 95, 97, 101, 103, 116, 118, 119, 120, 126, 132, 144, 146, 148, 151, 154, 172, 173, 177, 179, 185

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

The author's conjecture in A303401 has the following equivalent version: Each integer n > 1 can be written as the sum of two terms of the current sequence.

This has been verified for all n = 2..7*10^6.

			a(1) = 1 with 1 = 0*(3*0-1)/2 + 3^0.
a(2) = 2 with 2 = 1*(3*1-1)/2 + 3^0.
a(5) = 6 with 6 = 2*(3*2-1)/2 + 3^0.
a(6) = 8 with 8 = 2*(3*2-1)/2 + 3^1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 1367-1396.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000244, A000326, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432.

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0||Mod[Sqrt[24n+1]+1,6]==0);
tab={};Do[Do[If[PenQ[m-3^k],n=n+1;tab=Append[tab,m];Goto[aa]],{k,0,Log[3,m]}];Label[aa],{m,1,185}];Print[tab]

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A303233 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

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A303363 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.

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A303389 Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 5^c + 5^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

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A303393 Numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.

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A303399 Number of ordered pairs (a, b) with 0 <= a <= b such that n - 5^a - 5^b can be written as the sum of two triangular numbers.

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A303401 Number of ways to write n as a*(3*a-1)/2 + b*(3*b-1)/2 + 3^c + 3^d with a,b,c,d nonnegative integers.

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A303432 Number of ways to write n as a*(2*a-1) + b*(2*b-1) + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

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A303540 Number of ways to write n as a^2 + b^2 + binomial(2*c,c) + binomial(2*d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

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A303233 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

A303363 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2|c.

A303389 Number of ways to write n as a(a+1)/2 + b(b+1)/2 + 5^c + 5^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

A303401 Number of ways to write n as a(3a-1)/2 + b(3b-1)/2 + 3^c + 3^d with a,b,c,d nonnegative integers.

A303432 Number of ways to write n as a(2a-1) + b(2b-1) + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.

A303540 Number of ways to write n as a^2 + b^2 + binomial(2c,c) + binomial(2d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

A303434 Numbers of the form x(3x-1)/2 + 3^y with x and y nonnegative integers.