

A308411


Number of ways to write n as 2^i*3^j + A008347(k), where i, j and k > 0 are nonnegative integers.


6



0, 1, 2, 3, 4, 4, 4, 4, 5, 5, 5, 3, 5, 5, 4, 5, 5, 3, 6, 7, 6, 6, 5, 6, 7, 4, 5, 8, 6, 8, 7, 5, 5, 8, 8, 6, 7, 5, 7, 8, 7, 6, 6, 6, 8, 6, 5, 8, 6, 7, 7, 5, 8, 7, 8, 10, 9, 9, 6, 7, 8, 8, 6, 7, 10, 8, 8, 6, 9, 6, 7, 9, 8, 7, 7, 8, 9, 4, 8, 11, 10, 6, 9, 7, 11, 8, 8, 10, 9, 6, 4, 7, 10, 7, 7, 3, 11, 9, 10, 7
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OFFSET

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 1.
We have verified this for all n = 2..10^6.
QingHu Hou at Tianjin University extended the verification to 2*10^7. Then I used Hou's program to verify a(n) > 0 for n up to 10^9.  ZhiWei Sun, May 28 2019
Conjecture verified for n up to 10^10.  Giovanni Resta, May 28 2019


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 27942812.


EXAMPLE

a(2) = 1 with 2 = 2^0*3^0 + A008347(2).
a(3) = 2 with 3 = 2^0*3^0 + A008347(1) = 2^1*3^0 + A008347(2).


MATHEMATICA

f[n_]:=f[n]=FactorInteger[n];
FQ[n_]:=FQ[n]=n>0&&Part[f[n], Length[f[n]]][[1]]<4;
s[0]=0; s[n_]:=s[n]=Prime[n]s[n1];
tab={}; Do[r=0; Do[If[FQ[ns[k]], r=r+1], {k, 1, 2n1}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]


CROSSREFS

Cf. A000079, A000244, A008347, A308403, A316141.
Sequence in context: A195850 A253593 A174901 * A087848 A087844 A287635
Adjacent sequences: A308408 A308409 A308410 * A308412 A308413 A308414


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 25 2019


STATUS

approved



