A209254 Number of ways to write 2n-1 = p+q with q practical, p and p^4+q^4 both prime.

0, 1, 1, 2, 2, 2, 3, 1, 2, 2, 3, 2, 4, 3, 1, 3, 1, 1, 4, 2, 5, 5, 1, 4, 1, 2, 4, 3, 1, 6, 3, 4, 4, 5, 1, 6, 7, 2, 4, 3, 4, 2, 4, 5, 1, 2, 3, 7, 5, 2, 4, 8, 4, 6, 5, 1, 2, 2, 3, 8, 3, 1, 5, 6, 2, 4, 7, 4, 8, 4, 2, 7, 6, 3, 4, 3, 1, 6, 6, 1, 7, 6, 2, 8, 9, 5, 7, 3, 3, 10, 7, 3, 9, 14, 1, 9, 4, 3, 4, 6

Offset

1,4

Comments

Conjecture: a(n)>0 for all n>1.

Zhi-Wei Sun also conjectured that any odd integer greater than one can be written as p+q with q practical, and p and p^2+q^2 both prime. This is a refinement of Ming-Zhi Zhang's problem related to A036468.

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.

Example

a(8)=1 since 2*8-1=11+4 with 4 practical, 11 and 11^4+4^4=14897 both prime.

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[pr[2n-1-Prime[k]]==True&&PrimeQ[Prime[k]^4+(2n-1-Prime[k])^4]==True, 1, 0], {k, 1, PrimePi[2n-1]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

Cf. A000040, A005153, A208243, A208244, A208246, A209253.

Sequence in context: A141298 A256915 A348367 * A227738 A103960 A242626

Adjacent sequences: A209251 A209252 A209253 * A209255 A209256 A209257

Keyword

nonn

Author

Zhi-Wei Sun, Jan 14 2013

Status

approved