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A115156 Smallest number having exactly n ones in binary representation and also exactly n prime factors (counted with multiplicity). 2
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 (list; graph; refs; listen; history; text; internal format)
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 * A185072 A322507 A340471

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

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Last modified October 21 04:00 EDT 2021. Contains 348141 sequences. (Running on oeis4.)