This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129251 #42 Apr 18 2025 13:03:00
%S A129251 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,
%T A129251 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,
%U A129251 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
%N A129251 Number of distinct prime factors p of n such that p^p is a divisor of n.
%C A129251 Average value is A094289 = 0.28735...; attains record values on A076265, in particular a(A076265(n)) = n.
%H A129251 Reinhard Zumkeller, <a href="/A129251/b129251.txt">Table of n, a(n) for n = 1..10000</a>
%H A129251 Horst Alzer and Man Kam Kwong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Alzer/alzer15.html">On Sándor's Inequality for the Riemann Zeta Function</a>, J. Int. Seq. (2023) Vol. 26, Article 23.3.6.
%F A129251 a(A048103(n)) = 0, a(A100716(n)) > 0.
%F A129251 a(n) << sqrt(log n)/log log n. - _Charles R Greathouse IV_, Sep 14 2015
%F A129251 From _Antti Karttunen_, Aug 18 2016: (Start)
%F A129251 These formulas use Iverson bracket, which gives 1 as its value if the condition given inside [ ] is true and 0 otherwise:
%F A129251 a(1) = 0, for n > 1, a(n) = a(A028234(n)) + [A067029(n) >= A020639(n)].
%F A129251 Or, for n > 1, a(n) = a(A028234(n)) + [0 = n mod (A020639(n)^A020639(n))]. (End)
%F A129251 a(n) = Sum_{d|n} [rad(d) = Omega(d)*[omega(d) = 1]], where [ ] is the Iverson bracket. - _Wesley Ivan Hurt_, Feb 09 2022
%F A129251 Additive with a(p^e) = 1 if e >= p, and 0 otherwise. - _Amiram Eldar_, Nov 07 2022
%e A129251 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
%t A129251 {0}~Join~Table[Count[FactorInteger[n][[All, 1]], _?(Mod[n, #^#] == 0 &)], {n, 2, 120}] (* _Michael De Vlieger_, Oct 30 2019 *)
%o A129251 (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
%o A129251 (Scheme)
%o A129251 (define (A129251 n) (if (= 1 n) 0 (+ (A129251 (A028234 n)) (if (zero? (modulo n (expt (A020639 n) (A020639 n)))) 1 0))))
%o A129251 ;; _Antti Karttunen_, Aug 18 2016
%o A129251 (Scheme)
%o A129251 (define (A129251 n) (if (= 1 n) 0 (+ (A129251 (A028234 n)) (if (>= (A067029 n) (A020639 n)) 1 0))))
%o A129251 ;; _Antti Karttunen_, Aug 18 2016
%Y A129251 Cf. A129252, A020639, A028234, A001221, A051674, A067029, A094289.
%Y A129251 Cf. A048103 (indices of zeros), A100716 (nonzeros).
%Y A129251 Differs from A276077 for the first time at n=625, where a(625) = 0, while A276077(625) = 1.
%K A129251 nonn,easy
%O A129251 1,108
%A A129251 _Reinhard Zumkeller_, Apr 07 2007
%E A129251 Data section filled up to 120 terms by _Antti Karttunen_, Aug 18 2016