

A257922


Practical numbers m with m1 and m+1 both prime, and prime(m)1 and prime(m)+1 both practical.


2



4, 522, 1932, 5100, 6132, 6552, 8220, 18312, 18540, 22110, 29568, 45342, 70488, 70950, 92220, 105360, 109662, 114600, 116532, 117192, 123552, 128982, 131838, 132762, 136710, 148302, 149160, 166848, 177012, 183438, 197340, 206280, 233550, 235008, 257868, 272808, 273900, 276780, 279708, 286590
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OFFSET

1,1


COMMENTS

Conjecture: The sequence contains infinitely many terms. In other words, there are infinitely many positive integers n such that {prime(n)1, prime(n), prime(n)+1} is a "sandwich of the first kind" (A210479) and {n1, n, n+1} is a "sandwich of the second kind" (A258838).
This implies that there are infinitely many sandwiches of the first kind and also there are infinitely many sandwiches of the second kind.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.


EXAMPLE

a(1) = 4 since 4 is paractical with 41 and 4+1 twin prime, and prime(4)1 = 6 and prime(4)+1 = 8 are both practical.
a(2) = 522 since 522 is paractical with 5221 and 522+1 twin prime, and prime(522)1 = 3738 and prime(522)+1 = 3740 are both practical.


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<3Mod[n, 2]+Con[n]==0)
n=0; Do[If[PrimeQ[Prime[k]+2]&&pr[Prime[k]+1]&&pr[Prime[Prime[k]+1]1]&&pr[Prime[Prime[k]+1]+1], n=n+1; Print[n, " ", Prime[k]+1]], {k, 1, 24962}]


CROSSREFS

Cf. A000040, A005153, A014574, A209236, A210479, A257924, A258836, A258838, A259539.
Sequence in context: A291830 A003393 A089668 * A083284 A152218 A152463
Adjacent sequences: A257919 A257920 A257921 * A257923 A257924 A257925


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jul 12 2015


STATUS

approved



