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Least sum of n positive cubes to have exactly n prime factors, with multiplicity.
1

%I #22 Jan 31 2017 15:53:34

%S 9,66,56,108,144,192,256,512,1024,2048,4096,8192,16384,32768,65536,

%T 131072,262144,524288,1048576,2097152

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

%C 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).

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

%e a(2) = least semiprime in A003325 = 9 = 3 * 3 = 1^3 + 2^3 = A085366(1).

%e a(3) = least 3-almost prime in A003072 = 66 = 2 * 3 * 11 = 1^3 + 1^3 + 4^3 = A003072(10).

%e a(4) = least 4-almost prime in A003327 = 56 = 2^3 * 7 = 1^3 + 1^3 + 3^3 + 3^3 = A003327(10).

%e 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).

%e 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).

%p isSumcPosC := proc(n,c,minb)

%p local nrt ;

%p if c = 1 then nrt := iroot(n,3) ; if nrt^3 = n and n>= minb then true; else false; end if;

%p 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;

%p end proc:

%p A122733 := proc(n)

%p for a from 1 do if numtheory[bigomega](a) = n then if isSumcPosC(a,n,1) then return a; end if; end if;

%p end do:

%p end proc:

%p for n from 2 do print(A122733(n)) ; end do: # _R. J. Mathar_, Dec 22 2010

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

%K nonn

%O 2,1

%A _Jonathan Vos Post_, Sep 23 2006

%E a(17) from _Giovanni Resta_, Jun 13 2016

%E a(18)-a(21) more terms from _R. J. Mathar_, Jan 31 2017