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A211165 Number of ways to write n as the sum of a prime p with p-1 and p+1 both practical, a prime q with q+2 also prime, and a Fibonacci number. 2
0, 0, 0, 0, 0, 1, 1, 3, 3, 4, 5, 3, 5, 3, 4, 4, 3, 4, 4, 4, 6, 6, 8, 6, 8, 3, 7, 3, 6, 5, 5, 5, 7, 6, 11, 8, 12, 4, 8, 4, 7, 8, 6, 8, 8, 7, 11, 9, 13, 5, 8, 4, 7, 7, 6, 6, 6, 5, 7, 6, 10, 4, 9, 3, 9, 7, 8, 7, 6, 6, 7, 4, 7, 4, 7, 4, 8, 8, 11, 7, 6, 6, 8, 5, 6, 4, 7, 2, 9, 7, 12, 8, 7, 4, 10, 5, 9, 5, 8, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Conjecture: a(n)>0 for all n>5.

This has been verified for n up to 300000.

Note that for n=406 we cannot represent n in the given way with q+1 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, New Goldbach-type conjectures involving primes and practical numbers, a message to Number Theory List, Jan. 29, 2013.

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

EXAMPLE

a(6)=a(7)=1 since 6=3+3+0 and 7=3+3+1 with 3 and 5 both prime, 2 and 4 both practical, 0 and 1 Fibonacci numbers.

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)

pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True

q[n_]:=q[n]=PrimeQ[n]==True&&PrimeQ[n+2]==True

a[n_]:=a[n]=Sum[If[k!=2&&Fibonacci[k]<n&&pp[j]==True&&q[n-Fibonacci[k]-Prime[j]]==True, 1, 0], {k, 0, 2*Log[2, n]}, {j, 1, PrimePi[n-Fibonacci[k]]}]

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

CROSSREFS

Cf. A005153, A000045, A001359, A210479, A210681, A210722, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320, A210528, A210531, A210533.

Sequence in context: A258057 A003860 A108216 * A333868 A196210 A196477

Adjacent sequences:  A211162 A211163 A211164 * A211166 A211167 A211168

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 30 2013

STATUS

approved

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Last modified September 18 03:09 EDT 2021. Contains 347504 sequences. (Running on oeis4.)