A210528 Number of ways to write 2n = p+q (p<=q) with p, q and p^6+q^6 all practical.

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 3, 3, 3, 5, 3, 6, 2, 5, 4, 6, 3, 7, 3, 5, 4, 5, 3, 8, 3, 6, 6, 7, 4, 9, 4, 6, 6, 6, 3, 10, 4, 7, 8, 8, 3, 12, 4, 7, 9, 8, 4, 12, 5, 10, 8, 9, 4, 14, 3, 9, 8, 11, 4, 13, 4, 11, 9, 9, 4, 15, 4, 10, 9, 11, 5, 13, 4, 12, 11, 11, 5, 17, 4, 10, 11, 11, 4, 15, 4, 12, 11, 11, 3, 16, 3, 11, 12, 13

Offset: 1

Author

Zhi-Wei Sun, Jan 27 2013

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>0. Moreover, for any positive integers m and n, if m is greater than one or n is not among 17, 181, 211, 251, 313, 337, then 2n can be written as p+q with p, q and p^{3m}+q^{3m} all practical.

This conjecture implies that for each m=1,2,3,... there are infinitely many practical numbers of the form p^{3m}+q^{3m} with p and q both practical.

Examples

			a(3)=1 since 2*3=2+4 with 2, 4 and 2^6+4^6=4160 all 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, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Crossrefs

Cf. A005153, A002372, A208244, A208246, A209253, A209254, A209315, A209320.

Programs

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[k]==True&&pr[2n-k]==True&&pr[k^6+(2n-k)^6]==True,1,0],{k,1,n}]
Do[Print[n," ",a[n]],{n,1,100}]