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A261653 Number of primes p < n such that  n-p-1 and n+p+1 are both prime or both practical. 1
0, 0, 0, 0, 1, 0, 1, 2, 2, 3, 2, 3, 1, 2, 3, 5, 3, 3, 1, 4, 2, 4, 3, 5, 3, 3, 4, 4, 3, 4, 1, 3, 4, 5, 5, 7, 3, 1, 4, 6, 4, 7, 2, 4, 4, 5, 3, 8, 3, 4, 5, 6, 3, 6, 5, 6, 4, 4, 5, 9, 3, 2, 4, 7, 6, 10, 3, 6, 4, 6, 6, 10, 3, 3, 7, 7, 7, 9, 4, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Conjecture: a(n) > 0 for all n > 6. Also, for any integer n > 2, there is a prime p < n such that n-(p-1) and n+(p-1) are both prime or both practical.

Note that 1 is the only odd practical number and 2 is the only even prime.

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], 2012-2015.

EXAMPLE

a(31) = 1 since 11, 31-11-1 = 19 and 31+11+1 = 43 are all prime.

a(38) = 17 since 17 is prime, and 38-17-1 = 20 and 38+17+1 = 56 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)

p[n_]:=Prime[n]

Do[r=0; Do[If[(PrimeQ[n-p[k]-1]&&PrimeQ[n+p[k]+1])||(pr[n-p[k]-1]&&pr[n+p[k]+1]), r=r+1], {k, 1, PrimePi[n-1]}]; Print[n, " ", r]; Continue, {n, 1, 80}]

CROSSREFS

Cf. A000040, A005153, A208243, A209315, A209320, A209312, A261627, A261641.

Sequence in context: A279632 A230140 A156220 * A083900 A113517 A278399

Adjacent sequences:  A261650 A261651 A261652 * A261654 A261655 A261656

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 28 2015

STATUS

approved

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Last modified August 18 14:09 EDT 2017. Contains 290720 sequences.