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A258838 Practical numbers q with q-1 and q+1 twin primes: "Sandwiches of the second kind". 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

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.)

EXAMPLE

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.

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}]

CROSSREFS

Cf. A000040, A001359, A005153, A006512, A014574, A209236, A210452, A210479, A258836.

Sequence in context: A072570 A217259 A014574 * A034425 A073123 A079865

Adjacent sequences:  A258835 A258836 A258837 * A258839 A258840 A258841

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jun 12 2015

STATUS

approved

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Last modified April 20 16:17 EDT 2019. Contains 322310 sequences. (Running on oeis4.)