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

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, 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, 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

Offset

1,4

Links

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

Formula

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

a(A048103(n)) = 1.

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

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

Sequence in context: A268238 A081117 A368334 * A327936 A333748 A022929

Adjacent sequences: A129249 A129250 A129251 * A129253 A129254 A129255

Keyword

nonn

Author

Reinhard Zumkeller, Apr 07 2007

Extensions

Data section extended to a(120) by Antti Karttunen, Oct 01 2019

Status

approved