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a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.
5

%I #10 Jun 22 2022 02:32:55

%S 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,5,1,4,1,4,2,2,1,7,2,2,3,4,1,5,1,7,2,2,

%T 2,8,1,2,2,6,1,5,1,4,4,2,1,11,2,4,2,4,1,7,2,7,2,2,1,11,1,2,4,11,2,5,1,

%U 4,2,5,1,14,1,2,4,4,2,5,1,11,5,2,1,11,2

%N a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

%C First differs from A305254 at n = 40, from A001055 and A252665 at n = 36, from A218320 at n = 32 and from A317791, A318559 and A326334 at n = 30.

%H Amiram Eldar, <a href="/A355030/b355030.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) <= A001055(n).

%F a(p) = 1 for p prime.

%F a(A355031(n)) = n.

%e a(2) = 1 since numbers with 2 divisors are primes, i.e., numbers k with the single value Omega(k) = 1.

%e a(4) = 2 since numbers with 4 divisors are either of the following 2 forms: p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime.

%e a(8) = 3 since numbers with 8 divisors are either of the following 3 forms: p1 * p2 * p3 with p1, p2 and p3 being distinct primes, p1 * p2^3, or p1^7.

%t Table[Length[Union[Total[#-1]& /@ f[n]]], {n, 1, 100}] (* using the function f by _T. D. Noe_ at A162247 *)

%Y Row lengths of A355029.

%Y Cf. A000005, A001222, A118914, A162247, A355027.

%Y Cf. A001055, A218320, A305254, A317791, A318559, A326334.

%K nonn

%O 1,4

%A _Amiram Eldar_, Jun 16 2022