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

1,1

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

Links

Table of n, a(n) for n=1..27.

Example

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

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 *)

Crossrefs

Sequence in context: A303394 A074925 A141136 * A370749 A185072 A322507

Adjacent sequences: A115153 A115154 A115155 * A115157 A115158 A115159

Keyword

nonn

Author

Reinhard Zumkeller, Jan 14 2006

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

Status

approved