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A115561 a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

%I A115561 #26 Feb 16 2025 08:33:00
%S A115561 1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,2,1,3,1,5,1,1,1,2,1,1,3,7,1,5,1,2,1,1,
%T A115561 1,3,1,1,1,2,1,7,1,11,5,1,1,2,1,5,1,13,1,3,1,2,1,1,1,3,1,1,7,2,1,11,1,
%U A115561 17,1,7,1,2,1,1,5,19,1,13,1,2,3,1,1,3,1,1,1,2,1,3,1,23,1,1,1,2,1,7,11,5,1
%N A115561 a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.
%C A115561 From _Peter Munn_, Jul 14 2019: (Start)
%C A115561 a(n) = 1 if and only if n is 1 or a prime or semiprime. Otherwise a(n) is the 3rd factor when n is written as a product of primes in nondecreasing order. For example, 60 = 2*2*3*5, so a(60) = 3.
%C A115561 Although values equal to 1 are predominant at low indices, their asymptotic density is 0, whereas for values equal to prime(k) for k > 0 the asymptotic density is positive, namely A281890(k,3)/A002110(k)^3. For all sufficiently large n the median value of a(1), a(2), ... a(n) is A281889(3) = 433.
%C A115561 (End)
%H A115561 Antti Karttunen, <a href="/A115561/b115561.txt">Table of n, a(n) for n = 1..10000</a>
%H A115561 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LeastPrimeFactor.html">Least Prime Factor</a>
%F A115561 a(n) = A020639(A054576(n)).
%F A115561 If A001222(n) >= 3, a(n) = A027746(n,3), otherwise a(n) = 1. - _Peter Munn_, Jul 13 2019
%t A115561 f[n_] := FactorInteger[n][[1, 1]]; Table[f[#/f@ #] &[n/f@ n], {n, 101}] (* _Michael De Vlieger_, Aug 14 2017 *)
%o A115561 (Python)
%o A115561 from sympy import divisors, primefactors
%o A115561 def a032752(n): return 1 if n==1 else divisors(n)[-2]
%o A115561 def a020639(n): return 1 if n==1 else primefactors(n)[0]
%o A115561 def a(n): return a020639(a032752(a032752(n)))
%o A115561 print([a(n) for n in range(1, 102)]) # _Indranil Ghosh_, Aug 12 2017
%o A115561 (PARI) a020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1) \\ after _M. F. Hasler_ in A020639
%o A115561 a(n) = a020639((n/a020639(n))/a020639(n/a020639(n))) \\ _Felix Fröhlich_, Jul 15 2019
%Y A115561 Cf. A001222, A002110, A014673, A020639, A027746, A032742, A054576, A117358, A281889, A281890.
%K A115561 nonn
%O A115561 1,8
%A A115561 _Reinhard Zumkeller_, Mar 10 2006