|
%I #19 Mar 21 2015 04:48:11
%S 2,6,28,54,405,486,2808,4860,21870,40824,192456,524160,708588,4059072,
%T 14348907,58576608,123731712,462944160,1837080000,3874204890,
%U 11809800000,48183984000,65086642152,339033848832,1360965131136,2928898896840,6595446404736
%N Smallest number having exactly n ones in binary representation and also exactly n prime factors (counted with multiplicity).
%C A001222(a(n)) = A000120(a(n)) = n; subsequence of A071814.
%C a(n) is roughly 3^n and so far 4 <= a(n)/3^(n-2) <= 15. - _Robert G. Wilson v_
%C Does a(n) exist for every n? It exists for large enough n due to a result of Drmota, Mauduit, & Rivat, see A061712. _T. D. Noe_'s conjecture there implies that a(n) < 4*4^n. - _Charles R Greathouse IV_, Jul 30 2011
%e a(5) = 3*3*3*3*5 = 405_10 = 110010101_2.
%e a(10) = 2*2*2*3*3*3*3*3*3*7 = 40824_10 = 1001111101111000_2.
%e a(18) = 2*2*2*2*2*3*3*3*3*3*3*3*3*3*3*5*7*7 = 462944160_10 = 11011100101111111011110100000_2. - _Robert G. Wilson v_
%t Lk[n_] := Block[{k = 2^n - 1}, While[n != Plus @@ IntegerDigits[k, 2] || n != Plus @@ (Transpose[FactorInteger@k][[2]]), k++ ]; k]; L = {}; Do[v = Lk[n]; Print[{n, v}]; AppendTo[L, v], {n, 2, 16}]; L (Resta)
%t t = Table[0, {20}]; f[n_] := Block[{b = Count[ IntegerDigits[n, 2], 1], e = Plus @@ Last /@ FactorInteger@n}, If[b == e, b, 0]]; Do[ a = f@n; If[a > 0 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 550000000}]; t (* _Robert G. Wilson v_ *)
%t f[n_] := Min[ Select[ FromDigits[ #, 2] & /@ Permutations[ Join[ Table[0, {Max[6, 2n/3]}], Table[1, {n}]]], Plus @@ Last /@ FactorInteger@# == n &]]; Array[f, 18] (* _Robert G. Wilson v_ *)
%K nonn
%O 1,1
%A _Reinhard Zumkeller_, Jan 14 2006
%E a(14)-a(17) from _Giovanni Resta_, Jan 18 2006
%E a(14)-a(18) from _Robert G. Wilson v_, Jan 18 2006
%E a(19) from _Robert G. Wilson v_, Jan 22 2006
%E a(20)-a(24) from _Donovan Johnson_, Apr 07 2008
%E a(25)-a(27) from _Donovan Johnson_, Jul 30 2011
|
