A297167 a(1) = 0, for n > 1, a(n) = -1 + the excess of n (A046660) + the index of the largest prime factor (A061395).

0, 0, 1, 1, 2, 1, 3, 2, 2, 2, 4, 2, 5, 3, 2, 3, 6, 2, 7, 3, 3, 4, 8, 3, 3, 5, 3, 4, 9, 2, 10, 4, 4, 6, 3, 3, 11, 7, 5, 4, 12, 3, 13, 5, 3, 8, 14, 4, 4, 3, 6, 6, 15, 3, 4, 5, 7, 9, 16, 3, 17, 10, 4, 5, 5, 4, 18, 7, 8, 3, 19, 4, 20, 11, 3, 8, 4, 5, 21, 5, 4, 12, 22, 4, 6, 13, 9, 6, 23, 3, 5, 9, 10, 14, 7, 5, 24, 4, 5, 4, 25, 6, 26, 7, 3

Offset

1,5

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = A252464(n) - A001221(n).

For n > 1, a(n) = A033265(A156552(n)) = A297113(n) - 1.

For n > 1, a(n) = A046660(n) + A061395(n) - 1. - Antti Karttunen, Mar 13 2018

Mathematica

Array[-1 + PrimeOmega@ # - PrimeNu@ # + PrimePi[FactorInteger[#][[-1, 1]]] /. k_ /; k < 0 -> 0 &, 105] (* or, slightly faster *)
Array[-1 + Length@ # - Length@ Union@ # + PrimePi@ Last@ # /. k_ /; k < 0 -> 0 &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, #] &@ FactorInteger[#] &, 105] (* Michael De Vlieger, Mar 13 2018 *)

Prog

(PARI)
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A252464(n) = if(1==n, 0, (bigomega(n) + A061395(n) - 1));
A297167(n) = (A252464(n) - omega(n));
\\ Or just as:
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
\\ Antti Karttunen, Mar 13 2018
(Scheme) (define (A297167 n) (- (A252464 n) (A001221 n)))
(Python)
from sympy import factorint, primepi
def A297167(n): return primepi(max(f:=factorint(n)))+sum(e-1 for e in f.values())-1 if n>1 else 0 # Chai Wah Wu, Jul 29 2023

Crossrefs

Cf. A001221, A033265, A046660, A061395, A156552, A252464, A297113, A297168, A300827.

Sequence in context: A103151 A035221 A035191 * A325224 A303389 A276166

Adjacent sequences: A297164 A297165 A297166 * A297168 A297169 A297170

Keyword

nonn

Author

Antti Karttunen, Feb 27 2018

Extensions

Name changed, original equivalent definition is the first entry in the Formula section - Antti Karttunen, Mar 13 2018

Status

approved