A323022 Fourth omega of n. Number of distinct multiplicities in the prime signature of n.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Jan 02 2019

Keywords

• nonn

Comments

The indices of terms greater than 1 are {60, 84, 90, 120, 126, 132, 140, 150, ...}.

First term greater than 2 is a(1801800) = 3. In general, the first appearance of k is a(A182856(k)) = k.

The prime signature of n (row n of A118914) is the multiset of prime multiplicities in n.

We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first three omegas are A001222, A001221, A071625, and this sequence is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices).

Examples

			The prime signature of 1286485200 is {1, 1, 1, 2, 2, 3, 4}, in which 1 appears three times, two appears twice, and 3 and 4 both appear once, so there are 3 distinct multiplicities {1, 2, 3} and hence a(1286485200) = 3.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A001221, A001222, A006939, A025487, A056239, A059404, A062770, A071625, A118914, A181819, A182856, A182857, A304464, A304465, A323014, A323023.

Programs

Mathematica

red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
Table[PrimeNu[red[red[n]]],{n,200}]

PARI

a(n) = my(e=factor(n)[, 2], s = Set(e), m=Map(), v=vector(#s)); for(i=1, #s, mapput(m,s[i],i)); for(i=1, #e, v[mapget(m,e[i])]++); #Set(v) \\ David A. Corneth, Jan 02 2019

PARI

A071625(n) = #Set(factor(n)[, 2]); \\ From A071625
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A323022(n) = A071625(A181819(n)); \\ Antti Karttunen, Jan 03 2019

Formula

a(n) = A071625(A181819(n))

= A001221(A181819(A181819(n)))

= A001222(A181819(A181819(A181819(n))))

= A056239(A181819(A181819(A181819(A181819(n))))).

Extensions

More terms from Antti Karttunen, Jan 03 2019