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

4, 6, 18, 30, 198, 462, 1482, 2550, 3330, 4422, 9042, 11778, 26862, 38610, 47058, 60258, 62130, 65538, 69498, 79902, 96222, 106782, 124542, 143262, 149058, 151902, 184830, 200382, 208962, 225342, 237690, 249858, 251262, 295038, 301182, 312702, 345462, 348462

Offset: 1

Author

Zhi-Wei Sun, Jan 13 2013

Keywords

• nonn

Comments

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.

Note that this sequence is a subsequence of A014574.

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).

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

Examples

			a(3)=18 since {17,19} is a twin prime pair and 16, 18, 20 are practical numbers.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 2012-2017.

Crossrefs

Cf. A005153, A014574, A208243, A208244, A208246, A208249.

Programs

Mathematica

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

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