A229835 Number of ways to write n = (p - 1)/6 + q, where p is a prime, and q is a term of the sequence A000009.

0, 1, 2, 3, 3, 4, 5, 5, 5, 4, 6, 5, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 10, 9, 6, 8, 6, 10, 8, 9, 7, 7, 10, 10, 9, 8, 7, 10, 7, 10, 3, 7, 12, 8, 10, 6, 8, 9, 6, 10, 8, 11, 7, 11, 8, 7, 9, 8, 12, 10, 8, 12, 7, 9, 10, 10, 8, 11, 10, 7, 10, 9, 14, 9, 9, 9, 8, 10, 10, 9, 7, 8, 9, 9, 8, 10, 9, 10, 10, 9, 7, 8, 7, 12, 8

Offset

1,3

Comments

Conjecture: a(n) > 0 for all n > 1. Also, any integer n > 1 can be written as (p + 1)/6 + q, where p is a prime and q is a term of A000009.

We have verified this for n up to 2*10^8. Note that 26128189 cannot be written as (p - 1)/4 + q with p a prime and q a term of A000009. Also, 65152682 cannot be written as (p + 1)/4 + q with p a prime and q a term of A000009.

Links

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

Example

a(2) = 1 since 2 = (7 - 1)/ 6 + 1 with 7 prime, and 1 = A000009(i) for i = 0, 1, 2.
a(3) = 2 since 3 = (7 - 1 )/6 + 2 with 7 prime and 2 = A000009(3) = A000009(4), and 3 = (13 - 1 )/6 + 1 with 13 prime and 1 = A000009(i) for i = 0, 1, 2.

Mathematica

Do[m=0; Do[If[PartitionsQ[k]>=n, Goto[aa]]; If[k>1&&PartitionsQ[k]==PartitionsQ[k-1], Goto[bb]];
If[PrimeQ[6(n-PartitionsQ[k])+1], m=m+1]; Label[bb]; Continue, {k, 1, 2n}];
Label[aa]; Print[n, " ", m]; Continue, {n, 1, 100}]

Crossrefs

Cf. A000009, A000040, A232398, A232463, A233307, A233346, A233390, A233393, A233417, A233359.

Sequence in context: A286544 A238690 A332299 * A196155 A140858 A075458

Adjacent sequences: A229832 A229833 A229834 * A229836 A229837 A229838

Keyword

nonn

Author

Zhi-Wei Sun, Dec 19 2013

Status

approved