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

1,4

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.

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(3)=1 since 2*3=2+4 with 2, 4 and 2^6+4^6=4160 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<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}]

Crossrefs

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

Sequence in context: A278949 A030603 A092670 * A120450 A127238 A072114

Adjacent sequences: A210525 A210526 A210527 * A210529 A210530 A210531

Keyword

nonn

Author

Zhi-Wei Sun, Jan 27 2013

Status

approved