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%I #67 Feb 01 2022 14:34:58
%S 6,240,420,1032,1062,1278,2238,4020,12612,15972,19890,22110,34500,
%T 44772,134370,141768,145602,191142,217368,290658,436482,454578,464382,
%U 618030,668202,849348,888870,964260,1179150,1364970,1446900,1593498,1737102,1866438,2291802,3237432
%N Numbers m with m - 1, m + 1 and q(m) - 1 all prime, where q(.) is the strict partition function (A000009).
%C 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.
%C See A235357 for primes of the form q(m) - 1 with m - 1 and m + 1 both prime.
%C See also A235344 for a similar sequence.
%H Zhi-Wei Sun, <a href="/A235346/b235346.txt">Table of n, a(n) for n = 1..36</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;2a0328e6.1401">Twin primes and the strict partition function</a>, a message to Number Theory List, Jan. 15, 2014.
%e 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.
%t f[k_]:=PartitionsQ[Prime[k]+1]-1
%t n=0;Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[f[k]],n=n+1;Print[n," ",Prime[k]+1]],{k,1,10000}]
%t 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 *)
%Y Cf. A000009, A000040, A014574, A234530, A234569, A234644, A235343, A235344, A235356, A235357, A235358.
%K nonn,hard
%O 1,1
%A _Zhi-Wei Sun_, Jan 06 2014
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