A235344 Numbers m with m - 1, m + 1 and q(m) + 1 all prime, where q(.) is the strict partition function (A000009).

4, 6, 18, 42, 72, 102, 270, 282, 312, 618, 1032, 1062, 1320, 1950, 2082, 3528, 7350, 7488, 10332, 15138, 17388, 21600, 40038, 44700, 134922, 156258, 187908, 243708, 339138, 389568, 495360, 610920, 761712, 911292, 916218, 943800, 1013532, 1217472, 1312602

Offset

1,1

Comments

Clearly, any term after the first term 4 is a multiple of 6. By part (i) of the conjecture in A235343, this sequence should have infinitely many terms. The prime q(a(51)) + 1 = q(3235368) + 1 has 1412 decimal digits.

See A235356 for primes of the form q(m) + 1 with m - 1 and m + 1 both prime.

See also A235346 for a similar sequence.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..51

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

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(1) = 4 since 4 - 1, 4 + 1 and q(4) + 1 = 3 are all prime.
a(2) = 6 since 6 - 1, 6 + 1 and q(6) + 1 = 5 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}]

Crossrefs

Cf. A000009, A000040, A014574, A234530, A234569, A234644, A235343, A235346, A235356, A235357, A235358.

Sequence in context: A175955 A303526 A064403 * A060667 A218065 A005959

Adjacent sequences: A235341 A235342 A235343 * A235345 A235346 A235347

Keyword

nonn,hard

Author

Zhi-Wei Sun, Jan 06 2014

Status

approved