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A060687 Numbers n such that there exist exactly 2 Abelian groups of order n, i.e., A000688(n) = 2. 46

%I #33 Sep 16 2017 10:20:22

%S 4,9,12,18,20,25,28,44,45,49,50,52,60,63,68,75,76,84,90,92,98,99,116,

%T 117,121,124,126,132,140,147,148,150,153,156,164,169,171,172,175,188,

%U 198,204,207,212,220,228,234,236,242,244,245,260,261,268,275,276,279

%N Numbers n such that there exist exactly 2 Abelian groups of order n, i.e., A000688(n) = 2.

%C n belongs to this sequence iff exactly one prime in its factorization into prime powers has exponent 2 and all the other primes in the factorization have exponent 1, for example 60 = 2^2 * 3 * 5.

%C Numbers n such that A046660(n) = 1. - _Zak Seidov_, Nov 14 2012

%H Enrique Pérez Herrero, <a href="/A060687/b060687.txt">Table of n, a(n) for n = 1..5000</a>

%H Eckford Cohen, <a href="http://www.ams.org/journals/proc/1962-013-04/S0002-9939-1962-0138606-6/">Arithmetical notes. VIII. An asymptotic formula of Rényi</a>, Proc. Amer. Math. Soc. 13 (1962), pp. 536-539.

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F n such that A001222(n)-A001221(n) = 1.

%F Cohen proved that a(n) = kn + O(sqrt(n) log log n), where k = A013661/A179119 = 1/A271971 = 4.981178... - _Charles R Greathouse IV_, Aug 02 2016

%t Select[Range[500], PrimeOmega[#] - PrimeNu[#] == 1 &] (* _Harvey P. Dale_, Sep 08 2011 *)

%o (PARI) for(n=1,279,if(bigomega(n)-omega(n)==1,print1(n,",")))

%o (PARI) is(n)=factorback(factor(n)[,2])==2 \\ _Charles R Greathouse IV_, Sep 18 2015

%o (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), t=k^2; 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 (Haskell)

%o a060687 n = a060687_list !! (n-1)

%o a060687_list = filter ((== 1) . a046660) [1..]

%o -- _Reinhard Zumkeller_, Nov 29 2015

%Y Cf. A000688, A046660, A271971, A013661, A179119.

%K nonn

%O 1,1

%A Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001

%E Corrected and extended by _Vladeta Jovovic_, Jul 05 2001

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)