

A210528


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


6



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
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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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205210 [MR96i:11106].
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 20122017.


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<3Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[k]==True&&pr[2nk]==True&&pr[k^6+(2nk)^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

ZhiWei Sun, Jan 27 2013


STATUS

approved



