A036430 -id:A036430 - cp's OEIS frontend

A323305 Number of divisors of the number of prime factors of n counted with multiplicity.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 4, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

a(1) = 1 by convention.

First differs from A036430 at a(64) = 4, A036430(64) = 3.

Positions of 1's are 1 and the prime numbers A008578.
Positions of 2's are A063989.
Cf. A000005, A001222, A067028, A167175, A323295, A323350.

Mathematica
```
Array[Length@*Divisors@*PrimeOmega,100]
```

a(n) = if (n==1, 1, numdiv(bigomega(n))); \\ Michel Marcus, Jan 13 2019

a(n) = A000005(A001222(n)).

A073855 Number of steps to reach 0 starting with n and applying the process x ->bigomega(x), where bigomega = A001222.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 2, 4, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 3, 4, 3, 3, 2, 4, 2, 3, 3, 4, 3, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 3, 2, 3, 4, 3, 2, 4, 3, 3, 3, 4, 2, 4, 3, 3, 3, 3, 3, 4, 2, 3, 3, 4, 2, 3, 2, 4, 3

Offset: 0

Benoit Cloitre, Sep 02 2002

			bigomega(36) = 4, bigomega(4) = 2, bigomega(2) = 1, bigomega(1) = 0, hence a(36) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A001222, A036430.

A073855 := proc(n)
    option remember;
    if n <=0 then
        0;
    else
        1+procname(numtheory[bigomega](n)) ;
    end if;
end proc:
seq(A073855(n),n=0..20) ; # R. J. Mathar, Jul 31 2017

Table[-1 + Length@ NestWhileList[PrimeOmega, n, # > 0 &], {n, 0, 105}] (* Michael De Vlieger, Jul 29 2017 *)

a(n)=if(n<=0,0,s=n; c=1; while(bigomega(s)>0,s=bigomega(s); c++); c)

A073855(n) = if(!n,n,1+A073855(bigomega(n))); \\ Antti Karttunen, Jul 28 2017

first(n) = my(v = vector(n-1)); v[1] = 1; for(i=2, #v, v[i] = 1 + v[bigomega(i)]); concat([0], v) \\ David A. Corneth, Jul 28 2017

a(n) = 1+A036430(n).
For n >= 1, a(n) = 1 + a(bigomega(n)). - Vladeta Jovovic, Jul 10 2004
With a(0) = 0 as the termination condition of the recurrence. - Antti Karttunen, Jul 28 2017

More terms from Vladeta Jovovic, Jul 10 2004

Term a(0)=0 prepended by Antti Karttunen, Jul 28 2017

A287841 Number of iterations of number of distinct prime factors (A001221) needed to reach 1 starting at n (n is counted).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3

Offset: 1

Ilya Gutkovskiy, Jun 01 2017

nonn

			If n = 6 the trajectory is {6, 2, 1}. Its length is 3, thus a(6) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences computed from exponents in factorization of n

Cf. A001221, A036430, A036459, A049108, A073855, A115658 (first occurrence), A246655 (positions of 2).

f[n_] := Length[NestWhileList[ PrimeNu, n, # != 1 &]]; Array[f, 105]
a[1] = 1; a[n_] := a[n] = a[PrimeNu[n]] + 1; Table[a[n], {n, 105}]

A287841(n) = if(1==n,n,1+A287841(omega(n))); \\ Antti Karttunen, Nov 23 2017

from sympy import primefactors
def a(n): return 1 if n==1 else a(len(primefactors(n))) + 1 # Indranil Ghosh, Jun 03 2017

a(n) = a(omega(n)) + 1 for n > 1, where omega() is the number of distinct prime factors.

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A323305 Number of divisors of the number of prime factors of n counted with multiplicity.

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A073855 Number of steps to reach 0 starting with n and applying the process x ->bigomega(x), where bigomega = A001222.

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A287841 Number of iterations of number of distinct prime factors (A001221) needed to reach 1 starting at n (n is counted).

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