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A129251
Number of distinct prime factors p of n such that p^p is a divisor of n.
44
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
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. 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
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