A233390 a(n) = |{0 < k < n: 2^k - 1 + q(n-k) is prime}|, where q(.) is the strict partition function (A000009).

0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 4, 4, 2, 3, 2, 3, 3, 7, 4, 4, 5, 3, 4, 5, 5, 5, 6, 7, 6, 5, 4, 4, 9, 3, 6, 6, 5, 4, 7, 1, 4, 5, 6, 9, 6, 8, 6, 8, 4, 5, 8, 7, 4, 3, 4, 7, 6, 6, 3, 6, 5, 6, 4, 6, 8, 7, 8, 4, 5, 3, 6, 7, 7, 3, 10, 7, 5, 6, 10, 4, 8, 4, 6, 7, 6, 8, 10, 4, 6, 8, 9, 5, 6, 5, 7, 13, 5, 5, 6

Offset: 1

Author

Zhi-Wei Sun, Dec 08 2013

Keywords

• nonn

Comments

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

We have verified this for n up to 150000. For n = 124669, the least positive integer k with 2^k - 1 + q(n-k) prime is 13413.

Examples

			a(6) = 1 since 2^2 - 1 + q(4) = 3 + 2 = 5 is prime.
a(10) = 1 since 2^4 - 1 + q(6) = 15 + 4 = 19 is prime.
a(41) = 1 since 2^{16} - 1 + q(25) = 65535 + 142 = 65677 is prime.
a(127) = 1 since 2^{21} - 1 + q(106) = 2097151 + 728260 = 2825411 is prime.
a(153) = 1 since 2^{70} - 1 + q(83) = 1180591620717411303423 + 101698 = 1180591620717411405121 is prime.
a(164) = 1 since 2^{26} - 1 + q(138) = 67108863 + 8334326 = 75443189 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..8000
• Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2017.

Crossrefs

Cf. A000009, A000040, A000225, A233307, A233346, A233359, A233360, A233393, A233417, A233439.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[2^k-1+PartitionsQ[n-k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]