A257924 Primes p with p-1, p+1, prime(p)-1 and prime(p)+1 all practical.

3, 7, 31, 89, 199, 8009, 11551, 20129, 23549, 38609, 47501, 67231, 96221, 97001, 103409, 111871, 120473, 131071, 143261, 146681, 168869, 174761, 183091, 193951, 196181, 208279, 208961, 219727, 229769, 237691, 238519, 240641, 247759, 270271, 290249, 291101, 293201, 337039, 340577, 352831

Offset

1,1

Comments

Conjecture: The sequence contains infinitely many terms. In other words, there are infinitely many prime numbers p such that {p-1, p, p+1} and {prime(p)-1, prime(p), prime(p)+1} are both "sandwiches of the first kind" (A210479).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(1) = 3 since 3 is prime with 3-1, 3+1, prime(3)-1 = 4 and prime(3)+1 = 6 all practical.
a(3) = 31 since 31 is prime with 31-1, 31+1, prime(31)-1 = 126 and prime(31)+1 = 128 all 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<3||Mod[n, 2]+Con[n]==0)
n=0; Do[If[pr[Prime[k]-1]&&pr[Prime[k]+1]&&pr[Prime[Prime[k]]-1]&&pr[Prime[Prime[k]]+1], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 30201}]

Crossrefs

Cf. A000040, A005153, A209236, A210479, A257922, A258838, A259539.

Sequence in context: A226216 A244114 A072881 * A358341 A132153 A002357

Adjacent sequences: A257921 A257922 A257923 * A257925 A257926 A257927

Keyword

nonn

Author

Zhi-Wei Sun, Jul 13 2015

Status

approved