A129251 Number of distinct prime factors p of n such that p^p is a divisor of n.

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

Offset

1,108

Comments

Average value is A094289 = 0.28735...; attains record values on A076265, in particular a(A076265(n)) = n.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Horst Alzer and Man Kam Kwong, On Sándor's Inequality for the Riemann Zeta Function, J. Int. Seq. (2023) Vol. 26, Article 23.3.6.

Formula

a(A048103(n)) = 0, a(A100716(n)) > 0.

a(n) << sqrt(log n)/log log n. - Charles R Greathouse IV, Sep 14 2015

From Antti Karttunen, Aug 18 2016: (Start)

These formulas use Iverson bracket, which gives 1 as its value if the condition given inside [ ] is true and 0 otherwise:

a(1) = 0, for n > 1, a(n) = a(A028234(n)) + [A067029(n) >= A020639(n)].

Or, for n > 1, a(n) = a(A028234(n)) + [0 = n mod (A020639(n)^A020639(n))].

(End)

a(n) = Sum_{d|n} [rad(d) = Omega(d)*[omega(d) = 1]], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Feb 09 2022

Additive with a(p^e) = 1 if e >= p, and 0 otherwise. - Amiram Eldar, Nov 07 2022

Example

Since 15 = 3^1 * 5^1, a(15) = 0. But 16 = 2^4 is divisible by 2^2, so a(16) = 1. - Michael B. Porter, Aug 18 2016

Mathematica

{0}~Join~Table[Count[FactorInteger[n][[All, 1]], _?(Mod[n, #^#] == 0 &)], {n, 2, 120}] (* Michael De Vlieger, Oct 30 2019 *)

Prog

(PARI) a(n)=my(s, t, v); forprime(p=2, , v=valuation(n, p); if(v, n/=p^v; if(v>=p, s++), if(p^p>n, return(s)))) \\ Charles R Greathouse IV, Sep 14 2015
(Scheme, two variants)
(define (A129251 n) (if (= 1 n) 0 (+ (A129251 (A028234 n)) (if (zero? (modulo n (expt (A020639 n) (A020639 n)))) 1 0))))
(define (A129251 n) (if (= 1 n) 0 (+ (A129251 (A028234 n)) (if (>= (A067029 n) (A020639 n)) 1 0))))
;; Antti Karttunen, Aug 18 2016

Crossrefs

Cf. A129252, A020639, A028234, A001221, A051674, A067029, A094289.

Cf. A048103 (indices of zeros), A100716 (nonzeros).

Differs from A276077 for the first time at n=625, where a(625) = 0, while A276077(625) = 1.

Sequence in context: A133698 A219488 A370813 * A276077 A276935 A235127

Adjacent sequences: A129248 A129249 A129250 * A129252 A129253 A129254

Keyword

nonn,easy

Author

Reinhard Zumkeller, Apr 07 2007

Extensions

Data section filled up to 120 terms by Antti Karttunen, Aug 18 2016

Status

approved