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

1,4

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A115727 A115726 A086413 * A322832 A348389 A073453

Adjacent sequences: A069349 A069350 A069351 * A069353 A069354 A069355

Keyword

nonn

Author

Reinhard Zumkeller, Mar 18 2002

Extensions

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

Status

approved