OFFSET
1,3
COMMENTS
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..8192
FORMULA
a(1) = 0; for n > 1: a(n) = 1 + a(A252463(n)).
Other identities. For all n >= 1:
a(A000040(n)) = n.
a(A001248(n)) = n+1.
a(A030078(n)) = n+2.
And in general, a(prime(n)^k) = n+k-1.
a(A000079(n)) = n. [I.e., a(2^n) = n.]
For all n >= 2:
a(n) = A001222(n) + A061395(n) - 1 = A001222(n) + A252735(n) = A061395(n) + A252736(n) = 1 + A252735(n) + A252736(n).
a(n) = A325134(n) - 1. - Gus Wiseman, Apr 02 2019
From Antti Karttunen, Apr 14 2019: (Start)
a(1) = 0; for n > 1: a(n) = 1 + a(A253553(n)).
(End).
EXAMPLE
From Gus Wiseman, Apr 02 2019: (Start)
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the size of the inner lining of the integer partition with Heinz number n, which is also the size of the largest hook of the same partition. For example, the partition with Heinz number 715 is (6,5,3), with diagram
o o o o o o
o o o o o
o o o
which has inner lining
o o
o o o
o o o
and largest hook
o o o o o o
o
o
both of which have size 8, so a(715) = 8.
(End)
MATHEMATICA
Table[If[n==1, 1, PrimeOmega[n]+PrimePi[FactorInteger[n][[-1, 1]]]]-1, {n, 100}] (* Gus Wiseman, Apr 02 2019 *)
PROG
(Scheme, two different versions)
;; Memoization-macro definec can be found from Antti Karttunen's IntSeq-library
(PARI)
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
(Python)
from sympy import primepi, primeomega, primefactors
def A252464(n): return primeomega(n)+primepi(max(primefactors(n)))-1 if n>1 else 0 # Chai Wah Wu, Jul 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 20 2014
STATUS
approved