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

1,3

COMMENTS

Conjecture: a(n) > 0 for all n > 1.

We have verified this for all n = 2..10^6.

Qing-Hu 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. - Zhi-Wei Sun, May 28 2019

Conjecture verified for n up to 10^10. - Giovanni Resta, May 28 2019

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), no.8, 2794-2812.

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[n-1];

tab={}; Do[r=0; Do[If[FQ[n-s[k]], r=r+1], {k, 1, 2n-1}]; 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

Zhi-Wei Sun, May 25 2019

STATUS

approved

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Last modified November 27 12:23 EST 2020. Contains 338683 sequences. (Running on oeis4.)