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.

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

1,4

Comments

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.

Links

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.

Formula

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.

Mathematica

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]

Crossrefs

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

Sequence in context: A055718 A007302 A362028 * A327491 A099910 A213325

Adjacent sequences: A303398 A303399 A303400 * A303402 A303403 A303404

Keyword

nonn

Author

Zhi-Wei Sun, Apr 23 2018

Status

approved