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Distance between n and the next number with the same number of prime factors (counted with multiplicity).
3

%I #12 Nov 22 2024 11:03:20

%S 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,

%T 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,

%U 4,2,5,5,2,8,6,3,1,2,5,14,4,28

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

%C a(n) <= n/2, with equality when n is a power of 2. - _Robert Israel_, Nov 21 2024

%H Robert Israel, <a href="/A178139/b178139.txt">Table of n, a(n) for n = 2..10000</a>

%F {min d >0: A001222(n+d)= A001222(n)}. [_R. J. Mathar_, May 31 2010]

%e 2 is prime. The next prime is 3, and 3-2= 1 = a(2).

%e For n=3, the next prime is 5, 5-3 = 2 = a(3).

%e For n=4, the next number with 2 prime factors is 6, 6-4 =2 = a(4).

%p W:= map(numtheory:-bigomega, [$2..150]):

%p f:= t -> ListTools:-Search(W[t], W[t+1..-1]):

%p map(f, [$1..100]); # _Robert Israel_, Nov 21 2024

%t 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.

%Y Cf. A176884.

%K nonn

%O 2,2

%A _Daniel Tisdale_, May 20 2010

%E Offset set to 2, and more terms added by _R. J. Mathar_, May 31 2010