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A129252 Smallest prime factor p of n such that p^p is a divisor of n, a(n)=1 if no such factor exists.

%I A129252 #20 Feb 09 2025 18:10:24
%S A129252 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 A129252 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 A129252 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 A129252 Smallest prime factor p of n such that p^p is a divisor of n, a(n)=1 if no such factor exists.
%H A129252 Reinhard Zumkeller, <a href="/A129252/b129252.txt">Table of n, a(n) for n = 1..10000</a>
%F A129252 a(n) = 1 iff A129251(n) = 0.
%F A129252 a(A048103(n)) = 1.
%F A129252 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 A129252 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 A129252 Array[If[IntegerQ@ #, #, 1] &@ First@ SelectFirst[FactorInteger[#], #1 <= #2 & @@ # &] &, 120] (* _Michael De Vlieger_, Oct 01 2019 *)
%o A129252 (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
%o A129252 (PARI) A129252(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(p)); if(pp > n, return(1))); }; \\ _Antti Karttunen_, Feb 09 2025
%Y A129252 Cf. A020639, A048103 (positions of 1's), A008578, A051674, A129251, A359550, A368333, A380528.
%Y A129252 Differs from A327936 for the first time at n=108.
%K A129252 nonn
%O A129252 1,4
%A A129252 _Reinhard Zumkeller_, Apr 07 2007
%E A129252 Data section extended to a(120) by _Antti Karttunen_, Oct 01 2019