

A304943


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


3



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
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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

ZhiWei Sun, Table of n, a(n) for n = 1..100000
ZhiWei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341353.
ZhiWei 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. 279310. (See also arXiv:1211.1588 [math.NT], 20122017.)


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[n1]+f[n2]+f[n3];
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[nf[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: A106028 A105307 A281012 * A248597 A082071 A082065
Adjacent sequences: A304940 A304941 A304942 * A304944 A304945 A304946


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 22 2018


STATUS

approved



