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A233390 a(n) = |{0 < k < n: 2^k - 1 + q(n-k) is prime}|, where q(.) is the strict partition function (A000009). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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.

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.

EXAMPLE

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.

MATHEMATICA

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

Table[a[n], {n, 1, 100}]

CROSSREFS

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

Sequence in context: A096495 A276062 A324386 * A324114 A011776 A297791

Adjacent sequences:  A233387 A233388 A233389 * A233391 A233392 A233393

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Dec 08 2013

STATUS

approved

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Last modified August 13 02:09 EDT 2020. Contains 336441 sequences. (Running on oeis4.)