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A257922 Practical numbers m with m-1 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 (list; graph; refs; listen; history; text; internal format)
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 {n-1, 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

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 involving primes and quadratic forms, arXiv:1211.1588.

EXAMPLE

a(1) = 4 since 4 is paractical with 4-1 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 522-1 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<3||Mod[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 A350613 A152218

Adjacent sequences:  A257919 A257920 A257921 * A257923 A257924 A257925

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jul 12 2015

STATUS

approved

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Last modified October 4 11:47 EDT 2022. Contains 357239 sequences. (Running on oeis4.)