A304465 - OEIS

A304465

If n is prime, set a(n) = 1. Otherwise, start with the multiset of prime factors of n, and given a multiset take the multiset of its multiplicities. Repeating this until a multiset of size 1 is reached, set a(n) to the unique element of this multiset.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 6, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 4, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3

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OFFSET

1,4

COMMENTS

a(1) = 0 by convention.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 08 2018

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(p^n) = n where p is any prime number.

a(product of n distinct primes) = n.

a(1) = 0; and for n > 1, if n = prime^k, a(n) = k, otherwise, if n is squarefree [i.e., A001221(n) = A001222(n)], a(n) = A001221(n), otherwise a(n) = a(A181819(n)). - Antti Karttunen, Nov 08 2018

EXAMPLE

Starting with the multiset of prime factors of 2520 we have {2,2,2,3,3,5,7} -> {1,1,2,3} -> {1,1,2} -> {1,2} -> {1,1} -> {2}, so a(2520) = 2.

MATHEMATICA

Table[Switch[n, 1, 0, _?PrimeQ, 1, _, NestWhile[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]], Length[#]>1&]//First], {n, 100}]

PROG

(PARI)

A181819(n) = factorback(apply(e->prime(e), (factor(n)[, 2])));

A304465(n) = if(1==n, 0, my(t=isprimepower(n)); if(t, t, t=omega(n); if(bigomega(n)==t), t, A304465(A181819(n)))); \\ Antti Karttunen, Nov 08 2018

CROSSREFS