A000688 -id:A000688 - cp's OEIS frontend

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

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

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

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.

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

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

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

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

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

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

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

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

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

Corrected and extended by Vladeta Jovovic, Jul 05 2001

A188585 Moebius inversion of sequence A000688, the number of factorizations of n into prime powers greater than 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Marc Bogaerts, Apr 04 2011

Dirichlet convolution product of A000688 with the Moebius function.
It appears that a(n) is nonzero for n in A001694, the powerful numbers. - T. D. Noe, Apr 06 2011 [This is correct: a(n) > 0 if and only if n is in A001694. - Amiram Eldar, Jun 10 2025]
There is a similar sequence defined by b(n) = Product_{i} floor(e(i)/2) where n = Product_{p} p(i)^e(i) is the usual prime factorization, which differs from a(n) at n = 64, 128, 256, 512, 576, 729,.... - R. J. Mathar, Sep 18 2012 [This sequence is A365550. - Amiram Eldar, Jun 10 2025]
The number of unordered factorizations of n into 1 and prime powers p^e where p is prime and e >= 2 (A025475). - Amiram Eldar, Jun 10 2025

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000688, A001694, A002865, A008683, A025475, A365550.

mtrf:=function ( f, x )     # the Moebius inversion formula
    local  d;
    d := DivisorsInt( x );
    return Sum( d, function ( i )
            return f( i ) * MoebiusMu( (x / i) );
        end );
end;
nra:=function ( x )         # the number of Abelian groups of order x
    local  pp, ll;
    pp := PrimePowersInt( x );
    ll := [ 1 .. Size( pp ) / 2 ];
    return Product( List( 2 * ll, function ( i )
              return NrPartitions( pp[i] );
          end ) );
end;
a:=function ( n )
    return mtrf( nra, n );
end;

with(numtheory): with(combinat):
a:= n-> add(mobius(n/d) *mul(numbpart(i[2]),
        i=ifactors(d)[2]), d=divisors(n)):
seq(a(n), n=1..110);  # Alois P. Heinz, Apr 07 2011

MobiusTransform[a_List] := Module[{n = Length[a], b}, b = Table[0, {i, n}];Do[b[[i]] = Plus @@ (MoebiusMu[i/Divisors[i]] a[[Divisors[i]]]), {i, n}]; b]; A688[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; MobiusTransform[Array[A688, 100]] (* T. D. Noe, Apr 06 2011 *)
f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 10 2025 *)

a(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1), factor(n)[, 2])); \\ Amiram Eldar, Jun 10 2025

from math import prod
from sympy import partition, factorint
def A188585(n): return prod(partition(e)-partition(e-1) for e in factorint(n).values()) # Chai Wah Wu, Jun 10 2025

a(n) = Sum_{d|n} A008683(n/d) * A000688(d).
Dirichlet g.f.: Product_{k>=2} zeta(k*s). - Ilya Gutkovskiy, Nov 03 2020
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = Product_{k>=3} zeta(k/2) = 10.0301441966843566206076085895839492473559217336... - Vaclav Kotesovec, Apr 22 2025
Multiplicative with a(p^e) = A002865(e). - Amiram Eldar, Jun 10 2025

A188581 Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 7, 4, 4, 2, 8, 2, 4, 4, 12, 2, 8, 2, 8, 4, 4, 2, 14, 4, 4, 7, 8, 2, 8, 2, 19, 4, 4, 4, 16, 2, 4, 4, 14, 2, 8, 2, 8, 8, 4, 2, 24, 4, 8, 4, 8, 2, 14, 4, 14, 4, 4, 2, 16, 2, 4, 8, 30, 4, 8, 2, 8, 4, 8, 2, 28, 2, 4, 8, 8, 4, 8, 2, 24, 12, 4, 2, 16, 4, 4, 4, 14, 2, 16

Offset: 1

Marc Bogaerts, Apr 04 2011

			For n=8; the divisors of 8 are 1,2,4,8. There are 1,1,2,3 abelian groups of these orders respectively, so a(n) = 1+1+2+3 = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000688, A000070.

trf:=function ( f, x )  # the Dirichlet convolution 1 * f
    local  d;
    d := DivisorsInt( x );
    return Sum( d, function ( i )
            return f( i );
        end );
end;
nra:=function ( x )     # the number of Abelian Groups of order(n)
    local  pp, ll;
    pp := PrimePowersInt( x );
    ll := [ 1 .. Size( pp ) / 2 ];
    return Product( List( 2 * ll, function ( i )
              return NrPartitions( pp[i] );
          end ) );
end;
a:=function ( n )
    return trf( nra, n );
end;

with(combinat): with(numtheory):
a:= n-> add(mul(numbpart(i[2]), i=ifactors(d)[2]), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 08 2011

InverseMobiusTransform[a_List] := Module[{n = Length[a], b}, b = Table[0, {i, n}]; Do[b[[i]] = Plus @@ a[[Divisors[i]]], {i, n}]; b]; A688[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; InverseMobiusTransform[Array[A688, 100]] (* T. D. Noe, Apr 07 2011 *)
f[0] = 1; f[e_] := f[e] = f[e - 1] + PartitionsP[e]; a[1] = 1; a[n_] := Times @@ (f[Last[#]] & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 09 2020 *)

A000688(n)={local(f); f=factor(n); prod(i=1, matsize(f)[1], numbpart(f[i, 2]))}
A188581(n)=sumdiv(n,d,A000688(d))
r=vector(66,n,A188581(n)) /* show terms */ /* Joerg Arndt, Apr 08 2011 */

a(n) = Sum_{d | n} A000688(d).
Multiplicative with a(p^e) = A000070(e). - Amiram Eldar, Sep 09 2020
Dirichlet g.f.: zeta(s)^2 * Product_{k>=2} zeta(k*s). - Ilya Gutkovskiy, Nov 03 2020
Sum_{k=1..n} a(k) ~ n*((log(n) + 2*gamma - 1)*f(1) + f'(1)), where f(1) = Product_{k>=2} zeta(k) = A021002 = 2.1955691982567064617939..., f'(1) = f(1) * Sum_{k>=2} k*zeta'(k)/zeta(k) = -5.0385164470942955610707128990779476296197... and gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 21 2021

A050360 Abelian groups (factorizations into prime powers >1) indexed by prime signatures. A000688(A025487).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 3, 1, 7, 4, 5, 2, 11, 6, 7, 3, 15, 10, 4, 11, 1, 9, 5, 22, 14, 6, 15, 2, 15, 7, 30, 22, 10, 22, 3, 21, 8, 11, 42, 9, 30, 4, 25, 14, 30, 5, 33, 12, 15, 56, 15, 44, 1, 6, 35, 22, 42, 7, 45, 20, 22, 77, 21, 60, 2, 10, 55, 18, 30, 56, 8, 25, 11, 66, 28, 9, 30, 49, 101

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to groups

Cf. A000041, A000688, A000961, A050361-A050364.

Cf. A025487.

a050360 = a000688 . a025487  -- Reinhard Zumkeller, Aug 28 2014

More terms from David Wasserman, Feb 14 2002

A129667 Dirichlet inverse of the Abelian group count (A000688).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, 0, -1, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, 1, 1, -1, 0, -1, 1, 0, 1, -1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, 0, -1, -1, -1, 1, 1, 1, -1, 0, -1, 1, 1, 1, -1, 0, 1, 0, 1, 1, -1, -1, -1, 1, 1, 0, 1, -1, -1, 1, 1, -1, -1, 0, -1, 1, 1, 1, 1, -1, -1, 0, 0, 1, -1, -1, 1, 1, 1, 0, -1, -1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 0, -1, 1, -1

Offset: 1

Gerard P. Michon, Apr 28 2007, May 01 2007

The simple formula which gives the value of this multiplicative function for the power of any prime can be derived from Euler's celebrated "Pentagonal Number Theorem" (applied to the generating function of the partition function A000041 on which A000688 is based).

			a(8) and a(27) are zero because the sequence vanishes for the cubes of primes. Not so with fifth powers of primes (since 5 is a pentagonal number) so a(32) is nonzero.

R. J. Mathar, Table of n, a(n) for n = 1..1000
Gérard P. Michon, Multiplicative Functions.
Gérard P. Michon, Partition Function and Pentagonal Numbers.

Cf. A000041, A000326, A000688, A005449, A023900, A101035.

A000326inv := proc(n)
    local x,a ;
    for x from 0 do
        a := x*(3*x-1)/2 ;
        if a > n then
            return -1 ;
        elif a = n then
            return x;
        end if;
    end do:
end proc:
A005449inv := proc(n)
    local x,a ;
    for x from 0 do
        a := x*(3*x+1)/2 ;
        if a > n then
            return -1 ;
        elif a = n then
            return x;
        end if;
    end do:
end proc:
A129667 := proc(n)
    local a,e1,e2 ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        e1 := A000326inv(op(2,pe)) ;
        e2 := A005449inv(op(2,pe)) ;
        if e1 >= 0 then
            a := a*(-1)^e1 ;
        elif e2 >= 0 then
            a := a*(-1)^e2 ;
        else
            a := 0 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Nov 24 2017

a[n_] := a[n] = If[n == 1, 1, -Sum[FiniteAbelianGroupCount[n/d] a[d], {d, Most @ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 16 2020 *)

Multiplicative function for which a(p^e) either vanishes or is equal to (-1)^m, for any prime p, if e is either m(3m-1)/2 or m(3m+1)/2 (these integers are the pentagonal numbers of the first and second kind, A000326 and A005449).
Dirichlet g.f.: 1 / Product_{k>=1} zeta(k*s). - Ilya Gutkovskiy, Nov 06 2020
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Product_{p prime} ((1-1/p) * (1 + Sum_{m>=1} (1/p^(m*(3*m-1)/2) + 1/p^(m*(3*m+1)/2)))) = 0.85358290653064143678... . - Amiram Eldar, Feb 17 2024

A084911 Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) = ~B*n + ...

Original entry on oeis.org

7, 5, 2, 0, 1, 0, 7, 4, 2, 3, 7, 7, 0, 2, 9, 1, 6, 1, 5, 2, 0, 6, 3, 6, 0, 7, 7, 4, 5, 5, 4, 3, 2, 5, 7, 6, 5, 6, 0, 7, 1, 8, 1, 4, 6, 9, 5, 9, 1, 2, 8, 5, 2, 6, 6, 9, 6, 3, 9, 9, 7, 9, 8, 3, 2, 6, 7, 2, 3, 5, 0, 5, 6, 8, 4, 6, 4, 7, 9, 7, 3, 7, 8, 6, 3, 9, 4, 7, 3, 6, 3, 7, 8, 0, 8, 6, 5, 4, 3, 7, 1, 0, 1, 3, 6

Offset: 0

Eric W. Weisstein, Jun 11 2003

			0.7520107423...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See p. 16.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1; arXiv preprint, arXiv:1608.00795 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Abelian Group.

Cf. A000041, A000688, A021002, A084892, A084893, A272339.

digits = 10; m0 (* initial number of primes *) = 10^6; dm = 2*10^5; PP = PartitionsP; DP[n_] := DP[n] = (1/PP[n - 1] - 1 /PP[n]) // N[#, digits + 5]&; pmax = Prime[1000];
nmax[p_ /; p <= pmax] := nmax[p] = Module[{n}, For[n = 2, n < 1000, n++, If[Abs[1/PP[n - 1] - 1 /PP[n]]/p^n < 10^-100, Return[n]]]]; nmax[p_ /; p > pmax] := nmax[pmax];
s[p_] := Sum[DP[n]/p^n, {n, 2, nmax[p]}] ;
f[m_] := f[m] = Product[1 - s[p], {p, Prime[Range[m]]}]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 2][[1]] != RealDigits[f[m - dm], 10, digits + 2][[1]], m = m + dm; Print[m, " ", RealDigits[f[m]]]];
A0 = f[m]; RealDigits[A0, 10, digits][[1]] (* Jean-François Alcover, Apr 29 2016 *)

default(realprecision, 120); default(parisize, 10000000);
prodeulerrat((1-1/p)*(1 + sum(i = 1, 512, 1/(numbpart(i)*p^i)))) \\ Amiram Eldar, Mar 08 2024

Equals Product_{p prime} (1-Sum_{k >= 2} (1/P(k-1)-1/P(k))/p^k), where P(k) is the unrestricted partition function. - Jean-François Alcover, Apr 29 2016, [typo corrected by Vaclav Kotesovec, Mar 05 2024]

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} 1/A000688(k). - Amiram Eldar, Oct 16 2020

Data corrected by Jean-François Alcover, Apr 29 2016
a(10) from Vaclav Kotesovec, Mar 07 2024
More terms from Amiram Eldar, Mar 08 2024

A369162 a(n) = A000688(A000688(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A364388 at n = 42.

The sums of the first 10^k terms, for k = 1, 2, ..., are 10, 102, 1024, 10285, 102988, 1030280, 10304021, 103043644, 1030448091, 10304515936, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 1.0304... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, page 477-478.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, On the iterates of the enumerating function of finite Abelian groups, Bull. Cl. Sci. Math. Nat., Sci. Math., Vol. 17 (1989), pp. 13-22; alternative link.
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137. See p. 131, eq. 4.2.

Cf. A000688, A364388.
Cf. A048109, A059956, A271971.
Cf. A369163, A369164, A369165, A369167.

Table[FiniteAbelianGroupCount[FiniteAbelianGroupCount[n]], {n, 1, 100}]

A000688(n) = vecprod(apply(numbpart, factor(n)[, 2]));
a(n) = A000688(A000688(n));

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^4), where c = Sum_{k>=1} d(k) * A000688(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).

A369163 a(n) = A000005(A000688(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 36.

The sums of the first 10^k terms, for k = 1, 2, ..., are 13, 143, 1486, 15054, 151067, 1511982, 15123465, 151245456, 1512484372, 15124927227, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 1.512... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 73.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137. See p. 131, eq. 4.3.

Cf. A000005, A000688.
Cf. A048109, A059956, A271971.
Cf. A369162, A369164, A369165.
Cf. A007424, A278908, A307848, A323308, A358260, A365549.

Table[DivisorSigma[0, FiniteAbelianGroupCount[n]], {n, 1, 100}]

a(n) = numdiv(vecprod(apply(numbpart, factor(n)[, 2])));

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^4), where c = Sum_{k>=1} d(k) * A000005(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).

A370897 Partial alternating sums of the number of abelian groups sequence (A000688).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, 0, -3, -1, -2, -1, -3, -2, -3, -2, -7, -6, -8, -7, -9, -8, -9, -8, -11, -9, -10, -7, -9, -8, -9, -8, -15, -14, -15, -14, -18, -17, -18, -17, -20, -19, -20, -19, -21, -19, -20, -19, -24, -22, -24, -23, -25, -24, -27, -26, -29, -28, -29, -28

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000688, A063966.
Cf. A021002, A048651, A084892, A084893.
Similar sequences: A068762, A068773, A307704, A357817, A362028.

f[n_] := Times @@ (PartitionsP[Last[#]] & /@ FactorInteger[n]); f[1] = 1; Accumulate[Array[(-1)^(#+1) * f[#] &, 100]]

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * f(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A000688(k).

a(n) = k_1 * A021002 * n + k_2 * A084892 * n^(1/2) + k_3 * A084893 * n^(1/3) + O(n^(1/4 + eps)), where eps > 0 is arbitrarily small, k_j = -1 + 2 * Product_{i>=1} (1 - 1/2^(i/j)), k_1 = 2*A048651 - 1 = -0.422423809826..., k_2 = -0.924973966404..., and k_3 = -0.991478298912... (Tóth, 2017).

A369164 a(n) = A001221(A000688(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Jan 15 2024

First differs from A369165 at n = 36, from A080733 at n = 49, and from A107078 at n = 72.

The sums of the first 10^k terms, for k = 1, 2, ..., are 3, 40, 426, 4307, 43203, 432211, 4322486, 43226028, 432261887, 4322622387, ... . From these values the asymptotic mean of this sequence, whose existence was proven by Ivić (1983) (see the Formula section), can be empirically evaluated by 0.43226... .

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, page 164.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions, Journal of Number Theory, Vol. 16, No. 1 (1983), pp. 119-137. See p. 131, eq. 4.4.

Cf. A001221, A000688, A080733, A107078.
Cf. A048109, A059956, A271971.
Cf. A369162, A369163, A369165.

Table[PrimeNu[FiniteAbelianGroupCount[n]], {n, 1, 100}]

a(n) = omega(vecprod(apply(numbpart, factor(n)[, 2])));

Sum_{k=1..n} a(k) = c * n + O(sqrt(n) * log(n)^3/log(log(n))^2), where c = Sum_{k>=1} d(k) * A001221(k) is a constant, d(k) is the asymptotic density of the set {m | A000688(m) = k} (e.g., d(1) = A059956, d(2) = A271971, d(3) appears in A048109) (Ivić, 1983).

cp's OEIS Frontend

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

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A188585 Moebius inversion of sequence A000688, the number of factorizations of n into prime powers greater than 1.

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A188581 Inverse Moebius transform of A000688, the number of factorizations of n into prime powers greater than 1.

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A050360 Abelian groups (factorizations into prime powers >1) indexed by prime signatures. A000688(A025487).

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A129667 Dirichlet inverse of the Abelian group count (A000688).

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A084911 Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) = ~B*n + ...

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A369162 a(n) = A000688(A000688(n)).