

A210480


Number of primes p<n with p1 and p+1 both practical, and np prime or practical.


4



0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 3, 3, 4, 4, 4, 5, 4, 4, 3, 3, 2, 2, 4, 4, 4, 5, 5, 7, 6, 6, 3, 4, 3, 3, 5, 5, 4, 5, 5, 7, 7, 6, 4, 3, 3, 4, 4, 3, 2, 4, 4, 7, 6, 6, 3, 3, 4, 4, 4, 4, 2, 4, 4, 6, 5, 5, 3, 2, 4, 4, 6, 3, 3, 4, 4, 7, 5, 6, 4, 4, 4, 4, 7, 6, 5, 4, 3, 8, 5, 7, 3, 3, 5
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OFFSET

1,6


COMMENTS

Conjecture: a(n)>0 for all n>3.
This is stronger than Goldbach's conjecture and the author's conjecture that any odd number greater than one is the sum of a prime and a practical number. Also, it implies that there are infinitely many primes p with p1 and p+1 both practical.
The author has verified this new conjecture for n up to 10^7.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..60000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205210 [MR96i:11106].
ZhiWei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 20122017.


EXAMPLE

a(1846)=1 since 1846=1289+557 with 1289 and 557 both prime, and 1288 and 1290 both practical.
a(15675)=1 since 15675=919+14756 with 919 prime, and 918, 920, 14756 all 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<3Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[Prime[k]1]==True&&pr[Prime[k]+1]==True&&(PrimeQ[nPrime[k]]==Truepr[nPrime[k]]==True), 1, 0], {k, 1, PrimePi[n1]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]


CROSSREFS

Cf. A002372, A005153, A210479, A208243, A208244, A208246, A208249, A209236, A209253, A209254, A209312, A209315, A209320.
Sequence in context: A231717 A253315 A334138 * A321859 A321860 A266123
Adjacent sequences: A210477 A210478 A210479 * A210481 A210482 A210483


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 23 2013


STATUS

approved



