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A210452 Number of integers k<n with k-1 and k+1 both prime, and k and k*n both practical. 2
0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 3, 1, 3, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 4, 2, 5, 5, 4, 5, 5, 2, 4, 5, 5, 1, 5, 2, 6, 6, 5, 2, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 5, 2, 6, 3, 5, 7, 7, 7, 7, 3, 7, 7, 7, 3, 7, 4, 6, 8, 8, 8, 8, 3, 8, 8, 6, 3, 8, 8, 6, 8, 8, 3, 8, 8, 8, 7, 6, 8, 8, 3, 8, 8, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: a(n)>0 for all n>4.

This implies the twin prime conjecture since k*p is not practical for any prime p>sigma(k)+1.

Zhi-Wei Sun also made the following conjectures:

(1) For each integer n>197, there is a practical number k<n with k-2, k+2, k*n all practical.

(2) For every n=9,10,... there is a practical number k<n with k-4, k+4, k*n all practical.

(3) For any integer n>26863, the interval [1,n] contains five consecutive integers m-2, m-1, m, m+1, m+2 with m-1 and m+1 both prime, and m-2, m, m+2, m*n all practical.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

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

EXAMPLE

a(11)=1 since 5 and 7 are twin primes, and 6 and 6*11 are both practical.

MATHEMATICA

f[n_]:=f[n]=FactorInteger[n]

Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])

Con[n_]:=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_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)

a[n_]:=a[n]=Sum[If[PrimeQ[k-1]==True&&PrimeQ[k+1]==True&&pr[k]==True&&pr[k*n]==True, 1, 0], {k, 1, n-1}]

Do[Print[n, " ", a[n]], {n, 1, 100}]

CROSSREFS

Cf. A005153, A210444, A210445, A071558, A208243, A208244, A208246, A208249, A219185, A209253, A209254, A219312, A219315, A219320.

Sequence in context: A055020 A052435 A094701 * A240301 A289641 A209312

Adjacent sequences:  A210449 A210450 A210451 * A210453 A210454 A210455

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 20 2013

STATUS

approved

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Last modified August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)