This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A210480 #16 Dec 05 2018 06:13:21
%S A210480 0,0,0,1,1,2,2,3,3,3,3,2,2,2,2,2,1,3,3,4,4,4,5,4,4,3,3,2,2,4,4,4,5,5,
%T A210480 7,6,6,3,4,3,3,5,5,4,5,5,7,7,6,4,3,3,4,4,3,2,4,4,7,6,6,3,3,4,4,4,4,2,
%U A210480 4,4,6,5,5,3,2,4,4,6,3,3,4,4,7,5,6,4,4,4,4,7,6,5,4,3,8,5,7,3,3,5
%N A210480 Number of primes p<n with p-1 and p+1 both practical, and n-p prime or practical.
%C A210480 Conjecture: a(n)>0 for all n>3.
%C A210480 This is stronger than Goldbach's conjecture and the author's conjecture that any odd number greater than one is the sum of a prime and a practical number. Also, it implies that there are infinitely many primes p with p-1 and p+1 both practical.
%C A210480 The author has verified this new conjecture for n up to 10^7.
%H A210480 Zhi-Wei Sun, <a href="/A210480/b210480.txt">Table of n, a(n) for n = 1..60000</a>
%H A210480 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 A210480 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.
%H A210480 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 A210480 a(1846)=1 since 1846=1289+557 with 1289 and 557 both prime, and 1288 and 1290 both practical.
%e A210480 a(15675)=1 since 15675=919+14756 with 919 prime, and 918, 920, 14756 all practical.
%t A210480 f[n_]:=f[n]=FactorInteger[n]
%t A210480 Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
%t A210480 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 A210480 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
%t A210480 a[n_]:=a[n]=Sum[If[pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True&&(PrimeQ[n-Prime[k]]==True||pr[n-Prime[k]]==True),1,0],{k,1,PrimePi[n-1]}]
%t A210480 Do[Print[n," ",a[n]],{n,1,100}]
%Y A210480 Cf. A002372, A005153, A210479, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320.
%K A210480 nonn
%O A210480 1,6
%A A210480 _Zhi-Wei Sun_, Jan 23 2013