A129252 - OEIS

%I #14 Nov 07 2022 07:40:01

%S 1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,3,2,1,1,1,2,1,1,

%T 1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,3,1,2,1,1,1,2,1,1,1,2,1,1,1,2,

%U 1,1,1,2,1,1,1,2,1,1,1,2,3,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2

%N Smallest prime factor p of n such that p^p is a divisor of n, a(n)=1 if no such factor exists.

%H Reinhard Zumkeller, <a href="/A129252/b129252.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 iff A129251(n) = 0.

%F a(A048103(n)) = 1.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p^p) + Sum_{p prime} ((1/p^(p-1)) * Product_{primes q < p} (1-1/q^q)) = 1.30648526015949409005... . - _Amiram Eldar_, Nov 07 2022

%e For n = 108 = 2^2 * 3^3, it is 2 that is the smallest prime factor p satisfying p^p | 108, thus a(108) = 2.

%t Array[If[IntegerQ@ #, #, 1] &@ First@ SelectFirst[FactorInteger[#], #1 <= #2 & @@ # &] &, 120] (* _Michael De Vlieger_, Oct 01 2019 *)

%o (PARI) A129252(n) = { my(f = factor(n)); for(k=1, #f~, if(f[k, 2]>=f[k, 1], return(f[k, 1]))); (1); }; \\ _Antti Karttunen_, Oct 01 2019

%Y Cf. A020639, A048103, A008578, A051674, A129251.

%Y Differs from A327936 for the first time at n=108.

%K nonn

%O 1,4

%A _Reinhard Zumkeller_, Apr 07 2007

%E Data section extended to a(120) by _Antti Karttunen_, Oct 01 2019