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

Author

Zhi-Wei Sun, Jan 06 2014

Keywords

• nonn
• hard

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

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 *)