A115156 Smallest number having exactly n ones in binary representation and also exactly n prime factors (counted with multiplicity).

2, 6, 28, 54, 405, 486, 2808, 4860, 21870, 40824, 192456, 524160, 708588, 4059072, 14348907, 58576608, 123731712, 462944160, 1837080000, 3874204890, 11809800000, 48183984000, 65086642152, 339033848832, 1360965131136, 2928898896840, 6595446404736

Offset: 1

Author

Reinhard Zumkeller, Jan 14 2006

Keywords

• nonn

Comments

A001222(a(n)) = A000120(a(n)) = n; subsequence of A071814.

a(n) is roughly 3^n and so far 4 <= a(n)/3^(n-2) <= 15. - Robert G. Wilson v

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

Examples

			a(5) = 3*3*3*3*5 = 405_10 = 110010101_2.
a(10) = 2*2*2*3*3*3*3*3*3*7 = 40824_10 = 1001111101111000_2.
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_
		

Programs

Mathematica

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 = 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 *)
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 *)

Extensions

a(14)-a(17) from Giovanni Resta, Jan 18 2006

a(14)-a(18) from Robert G. Wilson v, Jan 18 2006

a(19) from Robert G. Wilson v, Jan 22 2006

a(20)-a(24) from Donovan Johnson, Apr 07 2008

a(25)-a(27) from Donovan Johnson, Jul 30 2011