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A235346 Numbers m with m - 1, m + 1 and q(m) - 1 all prime, where q(.) is the strict partition function (A000009). 7
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 27 12:49 EDT 2022. Contains 354097 sequences. (Running on oeis4.)