A122733 Least sum of n positive cubes to have exactly n prime factors, with multiplicity.

9, 66, 56, 108, 144, 192, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152

Offset: 2

Author

Jonathan Vos Post, Sep 23 2006

Keywords

• nonn

Comments

Sequence begins with n = 2 because a(1) is undefined (sum of one positive cube cannot have exactly one prime factor, i.e., be prime).

Examples

			a(2) = least semiprime in A003325 = 9 = 3 * 3 = 1^3 + 2^3 = A085366(1).
a(3) = least 3-almost prime in A003072 = 66 = 2 * 3 * 11 = 1^3 + 1^3 + 4^3 = A003072(10).
a(4) = least 4-almost prime in A003327 = 56 = 2^3 * 7 = 1^3 + 1^3 + 3^3 + 3^3 = A003327(10).
a(5) = least 5-almost prime in A003328 = 108 = 2^2 * 3^3 = 4^3 + 3^3 + 2^3 + 2^3 + 1^3 = A003328(25).
a(6) = least 6-almost prime in A003329 = 144 = 2^4 * 3^2 = 5^3 + 2^3 + 2^3 + 1^3 + 1^3 + 1^3 = A003329(46).
		

Crossrefs

Cf. A000578, A001222, A003072, A003325, A003327, A003328, A003329, A085366.

Programs

Maple

isSumcPosC := proc(n,c,minb)
        local nrt ;
        if c = 1 then nrt := iroot(n,3) ; if nrt^3 = n  and n>= minb then true; else false; end if;
        else for b from minb do if b^3 > n then return false; end if; if isSumcPosC(n-b^3,c-1,b) then return true; end if; end do: end if;
end proc:
A122733 := proc(n)
        for a from 1 do if numtheory[bigomega](a) = n then if isSumcPosC(a,n,1) then return a; end if; end if;
        end do:
end proc:
for n from 2 do print(A122733(n)) ; end do: # R. J. Mathar, Dec 22 2010

Formula

a(n) = Min{x = (c_1)^3 + (c_2)^3 + ... + (c_n)^3 such that omega(x) = A001222(x) = n}.

Extensions

a(17) from Giovanni Resta, Jun 13 2016

a(18)-a(21) more terms from R. J. Mathar, Jan 31 2017