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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A007306 A286544 A238690 * A196155 A140858 A075458

Adjacent sequences:  A229832 A229833 A229834 * A229836 A229837 A229838

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Dec 19 2013

STATUS

approved

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Last modified May 21 15:06 EDT 2019. Contains 323443 sequences. (Running on oeis4.)