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

2,2

Comments

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

Links

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

Formula

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

Example

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).

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.

Crossrefs

Cf. A176884.

Sequence in context: A275444 A243322 A071454 * A039642 A027300 A085313

Adjacent sequences: A178136 A178137 A178138 * A178140 A178141 A178142

Keyword

nonn

Author

Daniel Tisdale, May 20 2010

Extensions

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

Status

approved