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A190641 Numbers having exactly one non-unitary prime factor.

%I A190641 #41 Aug 02 2024 01:55:38
%S A190641 4,8,9,12,16,18,20,24,25,27,28,32,40,44,45,48,49,50,52,54,56,60,63,64,
%T A190641 68,75,76,80,81,84,88,90,92,96,98,99,104,112,116,117,120,121,124,125,
%U A190641 126,128,132,135,136,140,147,148,150,152,153,156,160,162,164
%N A190641 Numbers having exactly one non-unitary prime factor.
%C A190641 Numbers k such that the powerful part of k, A057521(k), is a composite prime power (A246547). - _Amiram Eldar_, Aug 01 2024
%H A190641 Reinhard Zumkeller, <a href="/A190641/b190641.txt">Table of n, a(n) for n = 1..10000</a>
%H A190641 Vaclav Kotesovec, <a href="/A190641/a190641.jpg">Graph - the asymptotic ratio</a>.
%H A190641 Carl Pomerance and Andrzej Schinzel, <a href="http://mjcnt.phystech.edu/en/article.php?id=4">Multiplicative Properties of Sets of Residues</a>, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.
%F A190641 A056170(a(n)) = 1.
%F A190641 a(n) ~ k*n, where k = Pi^2/(6*A154945) = 2.9816096.... - _Charles R Greathouse IV_, Aug 02 2016
%t A190641 Select[Range[164],Count[FactorInteger[#][[All, 2]], 1] == Length[FactorInteger[#]] - 1 &] (* _Geoffrey Critzer_, Feb 05 2015 *)
%o A190641 (Haskell)
%o A190641 a190641 n = a190641_list !! (n-1)
%o A190641 a190641_list = map (+ 1) $ elemIndices 1 a056170_list
%o A190641 (PARI) list(lim)=my(s=lim\4, v=List(), u=vectorsmall(s, i, 1), t, x); forprime(k=2, sqrtint(s), t=k^2; forstep(i=t, s, t, u[i]=0)); forprime(k=2, sqrtint(lim\1), for(e=2,logint(lim\1,k), t=k^e; for(i=1, #u, if(u[i] && gcd(k, i)==1, x=t*i; if(x>lim, break); listput(v, x))))); Set(v) \\ _Charles R Greathouse IV_, Aug 02 2016
%o A190641 (PARI) isok(n) = my(f=factor(n)); #select(x->(x>1), f[,2]) == 1; \\ _Michel Marcus_, Jul 30 2017
%Y A190641 Subsequence of A013929 and of A327877.
%Y A190641 Cf. A056170, A057521, A154945, A246547, A359466 (characteristic function).
%K A190641 nonn
%O A190641 1,1
%A A190641 _Reinhard Zumkeller_, Dec 29 2012