A178139 Distance between n and the next number with the same number of prime factors (counted with multiplicity).

1, 2, 2, 2, 3, 4, 4, 1, 4, 2, 6, 4, 1, 6, 8, 2, 2, 4, 7, 1, 3, 6, 12, 1, 7, 1, 2, 2, 12, 6, 16, 1, 1, 3, 4, 4, 1, 7, 14, 2, 2, 4, 1, 5, 3, 6, 24, 2, 2, 4, 11, 6, 2, 2, 4, 1, 4, 2, 21, 6, 3, 3, 32, 4, 2, 4, 2, 5, 5, 2, 8, 6, 3, 1, 2, 5, 14, 4, 28

Offset: 2

Author

Daniel Tisdale, May 20 2010

Keywords

• nonn

Comments

a(n) <= n/2, with equality when n is a power of 2. - Robert Israel, Nov 21 2024

Examples

			2 is prime. The next prime is 3, and 3-2 = 1 = a(2).
For n=3, the next prime is 5, 5-3 = 2 = a(3).
For n=4, the next number with 2 prime factors is 6, 6-4 = 2 = a(4).
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001222, A176884.

Programs

Maple

W:= map(numtheory:-bigomega, [$2..150]):
f:= t ->  ListTools:-Search(W[t], W[t+1..-1]):
map(f, [$1..100]); # Robert Israel, Nov 21 2024

Mathematica

a1=Array[Plus @@ Last /@ FactorInteger[ # ] &, 400];a2=Flatten[Position[a1, k]] (* In a2 putting 1,2,3,...for k gives table of positions of numbers with k factors, repetitions included. *)

Formula

a(n) = {min d > 0: A001222(n+d) = A001222(n)}. - R. J. Mathar, May 31 2010

Extensions

Offset set to 2, and more terms added by R. J. Mathar, May 31 2010