A261653 Number of primes p < n such that n-p-1 and n+p+1 are both prime or both practical.

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

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: A341417 A230140 A156220 * A347658 A343387 A083900

Adjacent sequences: A261650 A261651 A261652 * A261654 A261655 A261656

Keyword

nonn

Author

Zhi-Wei Sun, Aug 28 2015

Status

approved