cp's OEIS Frontend

A210452 Number of integers k with k-1 and k+1 both prime, and k and k*n both practical.

%I A210452 #16 Dec 08 2018 11:21:26
%S A210452 0,0,0,0,1,1,2,2,2,2,1,2,2,3,3,3,1,3,2,4,4,4,2,4,4,4,4,4,2,4,2,5,5,4,
%T A210452 5,5,2,4,5,5,1,5,2,6,6,5,2,6,6,6,6,6,2,6,6,6,6,5,2,6,3,5,7,7,7,7,3,7,
%U A210452 7,7,3,7,4,6,8,8,8,8,3,8,8,6,3,8,8,6,8,8,3,8,8,8,7,6,8,8,3,8,8,8
%N A210452 Number of integers k<n with k-1 and k+1 both prime, and k and k*n both practical.
%C A210452 Conjecture: a(n)>0 for all n>4.
%C A210452 This implies the twin prime conjecture since k*p is not practical for any prime p>sigma(k)+1.
%C A210452 Zhi-Wei Sun also made the following conjectures:
%C A210452 (1) For each integer n>197, there is a practical number k<n with k-2, k+2, k*n all practical.
%C A210452 (2) For every n=9,10,... there is a practical number k<n with k-4, k+4, k*n all practical.
%C A210452 (3) For any integer n>26863, the interval [1,n] contains five consecutive integers m-2, m-1, m, m+1, m+2 with m-1 and m+1 both prime, and m-2, m, m+2, m*n all practical.
%H A210452 Zhi-Wei Sun, <a href="/A210452/b210452.txt">Table of n, a(n) for n = 1..10000</a>
%H A210452 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 A210452 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.
%H A210452 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;20e70044.1301">Sandwiches with primes and practical numbers</a>, a message to Number Theory List, Jan. 13, 2013.
%e A210452 a(11)=1 since 5 and 7 are twin primes, and 6 and 6*11 are both practical.
%t A210452 f[n_]:=f[n]=FactorInteger[n]
%t A210452 Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
%t A210452 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}]
%t A210452 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
%t A210452 a[n_]:=a[n]=Sum[If[PrimeQ[k-1]==True&&PrimeQ[k+1]==True&&pr[k]==True&&pr[k*n]==True,1,0],{k,1,n-1}]
%t A210452 Do[Print[n," ",a[n]],{n,1,100}]
%Y A210452 Cf. A005153, A210444, A210445, A071558, A208243, A208244, A208246, A208249, A219185, A209253, A209254, A219312, A219315, A219320.
%K A210452 nonn
%O A210452 1,7
%A A210452 _Zhi-Wei Sun_, Jan 20 2013