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

1,1

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 * A377067 A034425 A073123

Adjacent sequences: A258835 A258836 A258837 * A258839 A258840 A258841

Keyword

nonn

Author

Zhi-Wei Sun, Jun 12 2015

Status

approved