A258838 Practical numbers q with q-1 and q+1 twin primes: "Sandwiches of the second kind".
4, 6, 12, 18, 30, 42, 60, 72, 108, 150, 180, 192, 198, 228, 240, 270, 312, 348, 420, 432, 462, 522, 570, 600, 660, 810, 828, 858, 882, 1020, 1032, 1050, 1092, 1152, 1230, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1620, 1722, 1872, 1932, 1950, 1998, 2028, 2088, 2112, 2130, 2142, 2268, 2310, 2340, 2550, 2592, 2688, 2730
Offset: 1
Keywords
Examples
a(1) = 4 since 4 is practical with 4-1 and 4+1 twin prime. a(2) = 6 since 6 is practical with 6-1 and 6+1 twin prime. a(3) = 12 since 12 is practical with 12-1 and 12+1 twin prime.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
- Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)
Programs
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Mathematica
f[n_]:=FactorInteger[n] Pow[n_,i_]:=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2]) 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_]:=n>0&&(n<3||Mod[n,2]+Con[n]==0) SW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&pr[n] n=0;Do[If[SW[m],n=n+1;Print[n," ",m]],{m,1,2730}]
Comments