cp's OEIS Frontend

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

%I A060687 #54 Aug 02 2025 09:59:10
%S A060687 4,9,12,18,20,25,28,44,45,49,50,52,60,63,68,75,76,84,90,92,98,99,116,
%T A060687 117,121,124,126,132,140,147,148,150,153,156,164,169,171,172,175,188,
%U A060687 198,204,207,212,220,228,234,236,242,244,245,260,261,268,275,276,279
%N A060687 Numbers k such that there exist exactly 2 Abelian groups of order k, i.e., A000688(k) = 2.
%C A060687 k 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 A060687 Numbers k such that A046660(k) = 1. - _Zak Seidov_, Nov 14 2012
%C A060687 Numbers that have twice as many unitary divisors as nonunitary divisors, the largest possible ratio for nonsquarefree numbers (i.e., numbers that have nonunitary divisors). - _Amiram Eldar_, Nov 01 2024
%H A060687 Amiram Eldar, <a href="/A060687/b060687.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from Enrique Pérez Herrero)
%H A060687 Eckford Cohen, <a href="https://doi.org/10.1090/S0002-9939-1962-0138606-6">Arithmetical notes. VIII. An asymptotic formula of Rényi</a>, Proc. Amer. Math. Soc. 13 (1962), pp. 536-539.
%H A060687 Alfred Rényi, <a href="http://elib.mi.sanu.ac.rs/files/journals/publ/14/13.pdf">On the density of certain sequences of integers</a>, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
%H A060687 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>.
%F A060687 k such that A001222(k)-A001221(k) = 1.
%F A060687 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 A060687 Select[Range[500], PrimeOmega[#] - PrimeNu[#] == 1 &] (* _Harvey P. Dale_, Sep 08 2011 *)
%o A060687 (PARI) for(n=1,279,if(bigomega(n)-omega(n)==1,print1(n,",")))
%o A060687 (PARI) is(n)=factorback(factor(n)[,2])==2 \\ _Charles R Greathouse IV_, Sep 18 2015
%o A060687 (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 A060687 (Haskell)
%o A060687 a060687 n = a060687_list !! (n-1)
%o A060687 a060687_list = filter ((== 1) . a046660) [1..]
%o A060687 -- _Reinhard Zumkeller_, Nov 29 2015
%o A060687 (Python)
%o A060687 from math import isqrt
%o A060687 from sympy import mobius, primerange
%o A060687 def A060687(n):
%o A060687     def bisection(f,kmin=0,kmax=1):
%o A060687         while f(kmax) > kmax: kmax <<= 1
%o A060687         kmin = kmax >> 1
%o A060687         while kmax-kmin > 1:
%o A060687             kmid = kmax+kmin>>1
%o A060687             if f(kmid) <= kmid:
%o A060687                 kmax = kmid
%o A060687             else:
%o A060687                 kmin = kmid
%o A060687         return kmax
%o A060687     def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A060687     def f(x): return int(n+x+sum(sum(g(x//p**j) if j&1 else -g(x//p**j) for j in range(2,x.bit_length())) for p in primerange(isqrt(x)+1)))
%o A060687     return bisection(f,n,n) # _Chai Wah Wu_, Feb 24 2025
%o A060687 (Python)
%o A060687 from sympy import factorint
%o A060687 def is_A060687(n): return sum(v := factorint(n).values()) == len(v) + 1 # _David Radcliffe_, Jul 28 2025
%Y A060687 Cf. A001221, A001222, A000688, A046660, A271971, A013661, A179119.
%K A060687 nonn,easy
%O A060687 1,1
%A A060687 Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
%E A060687 Corrected and extended by _Vladeta Jovovic_, Jul 05 2001