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

1,60

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).

Links

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

Formula

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

= A001221(A181819(A181819(n)))

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

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A298481 A324872 A307608 * A284562 A349542 A087102

Adjacent sequences: A323019 A323020 A323021 * A323023 A323024 A323025

Keyword

nonn

Author

Gus Wiseman, Jan 02 2019

Extensions

More terms from Antti Karttunen, Jan 03 2019

Status

approved