A308403 Number of ways to write n as 6^i + 3^j + A008347(k), where i, j and k > 0 are nonnegative integers.

0, 0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 2, 4, 3, 3, 4, 3, 2, 4, 2, 4, 5, 1, 3, 3, 2, 5, 4, 3, 6, 2, 4, 4, 4, 7, 4, 3, 3, 6, 7, 7, 3, 5, 3, 6, 7, 5, 7, 4, 4, 4, 5, 6, 7, 4, 4, 6, 6, 6, 6, 3, 6, 6, 6, 8, 7, 5, 3, 4, 6, 8, 4, 3, 4, 3, 6, 6, 4, 5, 6, 4, 6, 6, 9, 7, 4, 5, 8, 9, 6, 5, 5, 7, 5, 6, 2, 7, 6, 5

Offset

1,5

Comments

Conjecture 1: a(n) > 0 for all n > 2. In other words, each n = 3,4,... can be written as 6^i + 3^j + prime(k) - prime(k-1) + ... + (-1)^(k-1)*prime(1), where i, j and k > 0 are nonnegative integers.

Conjecture 2: If {a,b} is among {2,m} (m = 3..14), {3,4}, {3,5}, then any integer n > 2 can be written as a^i + b^j + A008347(k) with i, j and k > 0 nonnegative integers.

Using Qing-Hu Hou's program, we have verified Conjectures 1 and 2 for n up to 10^9 and 10^7 respectively. - Zhi-Wei Sun, May 28 2019

Conjecture 1 verified up to 10^10. Conjecture 2 holds up to 10^10 for all cases except {2, 12} since 4551086841 cannot be written as 2^i + 12^j + A008347(k). - 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(3) = 1 with 3 - (6^0 + 3^0) = 1 = A008347(2).
a(4) = 1 with 4 - (6^0 + 3^0) = 2 = A008347(1).
a(24) = 1 with 24 - (6^0 + 3^0) = 22 = A008347(13).
a(234) = 1 with 234 - (6^1 + 3^3) = 201 = A008347(90).
a(1134) = 1 with 1134 - (6^2 + 3^0) = 1097 = A008347(322).
a(4330) = 1 with 4330 - (6^3 + 3^0) = 4113 = A008347(1016).
a(5619) = 1 with 5619 - (6^1 + 3^3) = 5586 = A008347(1379).
a(6128) = 1 with 6128 - (6^0 + 3^0) = 6126 = A008347(1499).
a(16161) = 1 with 16161 - (6^3 + 3^0) = 15944 = A008347(3445).
a(133544) = 1 with 133544 - (6^0 + 3^8) = 126982 = A008347(22579).

Mathematica

Pow[n_]:=Pow[n]=n>0&&IntegerQ[Log[3, n]];
s[0]=0; s[n_]:=s[n]=Prime[n]-s[n-1];
tab={}; Do[r=0; Do[If[s[k]>=n, Goto[bb]]; Do[If[Pow[n-s[k]-6^m], r=r+1], {m, 0, Log[6, n-s[k]]}]; Label[bb], {k, 1, 2n-1}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000040, A000244, A000400, A008347, A303656, A303821, A308411.

Sequence in context: A105588 A064658 A194292 * A073578 A087866 A061392

Adjacent sequences: A308400 A308401 A308402 * A308404 A308405 A308406

Keyword

nonn

Author

Zhi-Wei Sun, May 25 2019

Status

approved