|
|
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.
|
|
27
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|