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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. 27
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

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.

EXAMPLE

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.

MATHEMATICA

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]

CROSSREFS

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

Sequence in context: A080748 A305374 A084966 * A224748 A165494 A204916

Adjacent sequences:  A303429 A303430 A303431 * A303433 A303434 A303435

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 23 2018

STATUS

approved

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Last modified February 18 19:54 EST 2019. Contains 320262 sequences. (Running on oeis4.)