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A209236 List of integers m>0 with m-1 and m+1 both prime, and m-2, m, m+2 all practical.

%I A209236 #28 May 03 2022 23:40:14
%S A209236 4,6,18,30,198,462,1482,2550,3330,4422,9042,11778,26862,38610,47058,
%T A209236 60258,62130,65538,69498,79902,96222,106782,124542,143262,149058,
%U A209236 151902,184830,200382,208962,225342,237690,249858,251262,295038,301182,312702,345462,348462
%N A209236 List of integers m>0 with m-1 and m+1 both prime, and m-2, m, m+2 all practical.
%C A209236 Conjecture: a(n) always exists. In other words, there are infinitely many quintuples (m-2, m-1, m, m+1, m+2) with m-1 and m+1 both prime and m-2, m, m+2 all practical.
%C A209236 Note that this sequence is a subsequence of A014574.
%C A209236 Zhi-Wei Sun observed that if m-2, m, m+2 are all practical with m>4 then m is congruent to 2 modulo 4. His PhD student Shan-Shan Du gave the following explanation: If m>4 is a multiple of 4, then m-2 and m+2 are congruent to 2 modulo 4, and one of them is not divisible by 3 and hence not practical (since 4=1+3).
%C A209236 Because all practical numbers greater than 2 are multiples of 4 or 6 (or both), it follows that every term in this sequence after the first is congruent to 6 modulo 12. - _Hal M. Switkay_, May 03 2022
%H A209236 Zhi-Wei Sun, <a href="/A209236/b209236.txt">Table of n, a(n) for n = 1..10000</a>
%H A209236 G. Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].
%H A209236 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arxiv:1211.1588 [math.NT], 2012-2017.
%e A209236 a(3)=18 since {17,19} is a twin prime pair and 16, 18, 20 are practical numbers.
%t A209236 f[n_] := f[n] = FactorInteger[n]; Pow[n_,i_] := Pow[n,i] = Part[Part[f[n],i],1]^(Part[Part[f[n],i],2]); Con[n_] := Con[n] = Sum[If[Part[Part[f[n],s+1],1] <= DivisorSigma[1, Product[Pow[n,i], {i,1,s}]] +1, 0, 1], {s,1,Length[f[n]]-1}]; pr[n_] := pr[n] = n>0 && (n<3 || Mod[n,2] + Con[n]==0); n=0; t = {}; Do[If[PrimeQ[Prime[k]+2] == True && pr[Prime[k]-1] == True && pr[Prime[k]+1] == True && pr[Prime[k]+3] == True, n = n+1; AppendTo[t, Prime[k]+1]], {k, 100}]; t
%o A209236 (PARI) o=3;forprime(p=5,,(2+o==o=p)||next; is_A005153(p-3) & is_A005153(p-1) & is_A005153(p+1) & print1(p-1,",")) \\ _M. F. Hasler_, Jan 13 2013
%Y A209236 Cf. A005153, A014574, A208243, A208244, A208246, A208249.
%K A209236 nonn
%O A209236 1,1
%A A209236 _Zhi-Wei Sun_, Jan 13 2013