A304943 Number of ways to write n as the sum of a positive tribonacci number (A000073) and a positive odd squarefree number.

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

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 6, 10, 11, 13, 29, 76, 1332, 25249.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000

Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.

Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Example

a(2) = 1 with 2 = 1 + 1, where 1 = A000073(2) = A000073(3) is a positive tribonacci number, and 1 is also odd and squarefree.
a(29) = 1 since 29 = A000073(8) + 5 with 5 odd and squarefree.
a(76) = 1 since 76 = A000073(6) + 3*23 with 3*23 odd and squarefree.
a(1332) = 1 since 1332 = A000073(7) + 1319 with 1319 odd and squarefree.
a(25249) = 1 since 25249 = A000073(4) + 25247 with 25247 odd and squarefree.

Mathematica

f[0]=0; f[1]=0; f[2]=1;
f[n_]:=f[n]=f[n-1]+f[n-2]+f[n-3];
QQ[n_]:=QQ[n]=Mod[n, 2]==1&&SquareFreeQ[n];
tab={}; Do[r=0; k=3; Label[bb]; If[f[k]>=n, Goto[aa]]; If[QQ[n-f[k]], r=r+1]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000073, A005117, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304720, A304721.

Sequence in context: A105307 A281012 A347342 * A248597 A082071 A082065

Adjacent sequences: A304940 A304941 A304942 * A304944 A304945 A304946

Keyword

nonn

Author

Zhi-Wei Sun, May 22 2018

Status

approved