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A235344
Numbers m with m - 1, m + 1 and q(m) + 1 all prime, where q(.) is the strict partition function (A000009).
7
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, 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}]
KEYWORD
nonn,hard
AUTHOR
Zhi-Wei Sun, Jan 06 2014
STATUS
approved