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

4, 9, 12, 18, 20, 25, 28, 44, 45, 49, 50, 52, 60, 63, 68, 75, 76, 84, 90, 92, 98, 99, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 188, 198, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268, 275, 276, 279

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

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.

Numbers k such that A046660(k) = 1. - Zak Seidov, Nov 14 2012

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Enrique Pérez Herrero)
• Eckford Cohen, Arithmetical notes. VIII. An asymptotic formula of Rényi, Proc. Amer. Math. Soc. 13 (1962), pp. 536-539.
• Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
• Index entries for sequences related to groups.

Crossrefs

Cf. A001221, A001222, A000688, A046660, A271971, A013661, A179119.

Programs

Haskell

a060687 n = a060687_list !! (n-1)
a060687_list = filter ((== 1) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Mathematica

Select[Range[500], PrimeOmega[#] - PrimeNu[#] == 1 &] (* Harvey P. Dale, Sep 08 2011 *)

PARI

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

PARI

is(n)=factorback(factor(n)[,2])==2 \\ Charles R Greathouse IV, Sep 18 2015

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

Python

from math import isqrt
from sympy import mobius, primerange
def A060687(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    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)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

Python

from sympy import factorint
def is_A060687(n): return sum(v := factorint(n).values()) == len(v) + 1 # David Radcliffe, Jul 28 2025

Formula

k such that A001222(k)-A001221(k) = 1.

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

Extensions

Corrected and extended by Vladeta Jovovic, Jul 05 2001