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A258838
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Practical numbers q with q-1 and q+1 twin primes: "Sandwiches of the second kind".
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7
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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
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OFFSET
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1,1
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COMMENTS
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The author introduced two kinds of "sandwiches" in 2013. The conjecture in A258836 essentially says that {a(m)/a(n): m,n = 1,2,3,...} coincides with the set of all positive rational numbers. This implies that the sequence contains infinitely many terms.
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LINKS
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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.)
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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