A235346 Numbers m with m - 1, m + 1 and q(m) - 1 all prime, where q(.) is the strict partition function (A000009).

6, 240, 420, 1032, 1062, 1278, 2238, 4020, 12612, 15972, 19890, 22110, 34500, 44772, 134370, 141768, 145602, 191142, 217368, 290658, 436482, 454578, 464382, 618030, 668202, 849348, 888870, 964260, 1179150, 1364970, 1446900, 1593498, 1737102, 1866438, 2291802, 3237432

Offset

1,1

Comments

Clearly, each term is a multiple of 6. By the conjecture in A235358 (which is part (ii) of the conjecture in A235343), this sequence should have infinitely many terms. q(a(36)) - 1 = q(3237432) - 1 is a prime having 1412 decimal digits.

See A235357 for primes of the form q(m) - 1 with m - 1 and m + 1 both prime.

See also A235344 for a similar sequence.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..36

Zhi-Wei Sun, Twin primes and the strict partition function, a message to Number Theory List, Jan. 15, 2014.

Example

a(1) = 6 since q(4) - 1 = 1 is not a prime, and 6 - 1, 6 + 1 and q(6) - 1 = 3 are all prime.

Mathematica

f[k_]:=PartitionsQ[Prime[k]+1]-1
n=0; Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[f[k]], n=n+1; Print[n, " ", Prime[k]+1]], {k, 1, 10000}]
Select[Mean/@Select[Partition[Prime[Range[10000]], 2, 1], #[[2]]-#[[1]] == 2&], PrimeQ[PartitionsQ[#]-1]&] (* The program generates the first 14 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Feb 01 2022 *)

Crossrefs

Cf. A000009, A000040, A014574, A234530, A234569, A234644, A235343, A235344, A235356, A235357, A235358.

Sequence in context: A266657 A145180 A256275 * A077231 A172965 A002022

Adjacent sequences: A235343 A235344 A235345 * A235347 A235348 A235349

Keyword

nonn,hard

Author

Zhi-Wei Sun, Jan 06 2014

Status

approved