A069352 Total number of prime factors of 3-smooth numbers.

0, 1, 1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 5, 4, 5, 4, 6, 5, 4, 6, 5, 7, 6, 5, 7, 6, 5, 8, 7, 6, 8, 7, 6, 9, 8, 7, 6, 9, 8, 7, 10, 9, 8, 7, 10, 9, 8, 11, 7, 10, 9, 8, 11, 10, 9, 12, 8, 11, 10, 9, 12, 8, 11, 10, 13, 9, 12, 11, 10, 13, 9, 12, 11, 14, 10, 13, 9, 12, 11, 14, 10, 13, 12, 15, 11

Offset: 1

Author

Reinhard Zumkeller, Mar 18 2002

Keywords

• nonn

Comments

a(n) = A001222(A003586(n));

a(n) = A022328(n) + A022329(n);

A086414(n) <= A086415(n) <= a(n).

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003586, A001222, A003586, A022328, A022329, A086414, A086415.

Programs

Haskell

a069352 = a001222 . a003586  -- Reinhard Zumkeller, May 16 2015

Mathematica

smoothNumbers[p_, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j-1]^Take[aa, j-1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; PrimeOmega /@ smoothNumbers[3, 10^5] (* Jean-François Alcover, Nov 11 2016 *)

Formula

a(n) = i+j for 3-smooth numbers n = 2^i*3^j (A003586).

a(n) = A001222(A033845(n))-2. - Enrique Pérez Herrero, Jan 04 2012

Extensions

Edited by N. J. A. Sloane, Oct 27 2008 at the suggestion of R. J. Mathar.