A035218 -id:A035218 - cp's OEIS frontend

A048272 Number of odd divisors of n minus number of even divisors of n.

1, 0, 2, -1, 2, 0, 2, -2, 3, 0, 2, -2, 2, 0, 4, -3, 2, 0, 2, -2, 4, 0, 2, -4, 3, 0, 4, -2, 2, 0, 2, -4, 4, 0, 4, -3, 2, 0, 4, -4, 2, 0, 2, -2, 6, 0, 2, -6, 3, 0, 4, -2, 2, 0, 4, -4, 4, 0, 2, -4, 2, 0, 6, -5, 4, 0, 2, -2, 4, 0, 2, -6, 2, 0, 6, -2, 4, 0, 2, -6, 5, 0, 2, -4, 4, 0, 4, -4, 2, 0, 4, -2, 4

Offset: 1

Adam Kertesz

abs(a(n)) = (1/2) * (number of pairs (i,j) satisfying n = i^2 - j^2 and -n <= i,j <= n). - Benoit Cloitre, Jun 14 2003
As A001227(n) is the number of ways to write n as the difference of 3-gonal numbers, a(n) describes the number of ways to write n as the difference of e-gonal numbers for e in {0,1,4,8}. If pe(n):=(1/2)*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then 4*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=1, 2*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e in {0,4} and for a(n) itself is a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=8. (Same for e=-1 see A035218.) - Volker Schmitt (clamsi(AT)gmx.net), Nov 09 2004
An argument by Gareth McCaughan suggests that the average of this sequence is log(2). - Hans Havermann, Feb 10 2013 [Supported by a graph. - Vaclav Kotesovec, Mar 01 2023]
From Keith F. Lynch, Jan 20 2024: (Start)
a(n) takes every possible integer value, positive, negative, and zero. Proof: For all nonnegative integers k, a(3^k) = 1+k, a(2^k) = 1-k.
a(n) takes every possible integer value except 1 and -1 infinitely many times. Proof: a(o^(k-1)) = k and a(4*o^(k-1)) = -k for all positive integers k and odd primes o, of which there are infinitely many.  a(n) = 0 iff n = 2 (mod 4).  a(n) = 1 iff n = 1.  a(n) = -1 iff n = 4.
a(n) takes prime value p only for n = o^(p-1), where o is any odd prime.
Terms have a simple pattern that repeats with a period of 4: Positive, zero, positive, negative.
(End)
Inverse Möbius transform of (-1)^(n+1). - Wesley Ivan Hurt, Jun 22 2024

			a(20) = -2 because 20 = 2^2*5^1 and (1-2)*(1+1) = -2.
G.f. = x + 2*x^3 - x^4 + 2*x^5 + 2*x^7 - 2*x^8 + 3*x^9 + 2*x^11 - 2*x^12 + ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), first formula.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 97, 7(ii).

T. D. Noe, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n, for n = 1..1000000
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_{o-e}(n).
Index entries for sequences mentioned by Glaisher.

Cf. A048298. A diagonal of A060184.
First differences of A059851.
Cf. A001227, A035218, A094572, A099774, A112329, A183063, A000005.
Indices of records: A053624 (highs), A369151 (lows).

a048272 n = a001227 n - a183063 n  -- Reinhard Zumkeller, Jan 21 2012

[&+[(-1)^(d+1):d in Divisors(n)] :n in [1..95] ]; // Marius A. Burtea, Aug 10 2019

add(x^n/(1+x^n), n=1..60): series(%,x,59);
A048272 := proc(n)
    local a;
    a := 1 ;
    for pfac in ifactors(n)[2] do
        if pfac[1] = 2 then
            a := a*(1-pfac[2]) ;
        else
            a := a*(pfac[2]+1) ;
        end if;
    end do:
    a ;
end proc: # Schmitt, sign corrected R. J. Mathar, Jun 18 2016
# alternative Maple program:
a:= n-> -add((-1)^d, d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 28 2018

Rest[ CoefficientList[ Series[ Sum[x^k/(1 - (-x)^k), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)
dif[n_]:=Module[{divs=Divisors[n]},Count[divs,?OddQ]-Count[ divs, ?EvenQ]]; Array[dif,100] (* Harvey P. Dale, Aug 21 2011 *)
a[n]:=Sum[-(-1)^d,{d,Divisors[n]}] (* Steven Foster Clark, May 04 2018 *)
f[p_, e_] := If[p == 2, 1 - e, 1 + e]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 09 2022 *)

{a(n) = if( n<1, 0, -sumdiv(n, d, (-1)^d))}; /* Michael Somos, Jul 22 2006 */

N=17; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j); \\ log case
s=-log(prod(j=1,N,(1+x^j)^(1/j)));
s=serconvol(s,c)
v=Vec(s) \\ Joerg Arndt, May 03 2008

a(n)=my(o=valuation(n,2),f=factor(n>>o)[,2]);(1-o)*prod(i=1,#f,f[i]+1) \\ Charles R Greathouse IV, Feb 10 2013

a(n)=direuler(p=1,n,if(p==2,(1-2*X)/(1-X)^2,1/(1-X)^2))[n] /* Ralf Stephan, Mar 27 2015 */

{a(n) = my(d = n -> if(frac(n), 0, numdiv(n))); if( n<1, 0, if( n%4, 1, -1) * (d(n) - 2*d(n/2) + 2*d(n/4)))}; /* Michael Somos, Aug 11 2017 */

Coefficients in expansion of Sum_{n >= 1} x^n/(1+x^n) = Sum_{n >= 1} (-1)^(n-1)*x^n/(1-x^n). Expand Sum 1/(1+x^n) in powers of 1/x.
If n = 2^p1*3^p2*5^p3*7^p4*11^p5*..., a(n) = (1-p1)*Product_{i>=2} (1+p_i).
Multiplicative with a(2^e) = 1 - e and a(p^e) = 1 + e if p > 2. - Vladeta Jovovic, Jan 27 2002
a(n) = (-1)*Sum_{d|n} (-1)^d. - Benoit Cloitre, May 12 2003
Moebius transform is period 2 sequence [1, -1, ...]. - Michael Somos, Jul 22 2006
G.f.: Sum_{k>0} -(-1)^k * x^(k^2) * (1 + x^(2*k)) / (1 - x^(2*k)) [Ramanujan]. - Michael Somos, Jul 22 2006
Equals A051731 * [1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Nov 07 2007
From Reinhard Zumkeller, Jan 21 2012: (Start)
a(n) = A001227(n) - A183063(n).
a(A008586(n)) < 0; a(A005843(n)) <= 0; a(A016825(n)) = 0; a(A042968(n)) >= 0; a(A005408(n)) > 0. (End)
a(n) = Sum_{k=0..n} A081362(k)*A015723(n-k). - Mircea Merca, Feb 26 2014
abs(a(n)) = A112329(n) = A094572(n) / 2. - Ray Chandler, Aug 23 2014
From Peter Bala, Jan 07 2015: (Start)
Logarithmic g.f.: log( Product_{n >= 1} (1 + x^n)^(1/n) ) = Sum_{n >= 1} a(n)*x^n/n.
a(n) = A001227(n) - A183063(n). By considering the logarithmic generating functions of these three sequences we obtain the identity
( Product_{n >= 0} (1 - x^(2*n+1))^(1/(2*n+1)) )^2 = Product_{n >= 1} ( (1 - x^n)/(1 + x^n) )^(1/n). (End)
Dirichlet g.f.: zeta(s)*eta(s) = zeta(s)^2*(1-2^(-s+1)). - Ralf Stephan, Mar 27 2015
a(2*n - 1) = A099774(n). - Michael Somos, Aug 12 2017
From Paul D. Hanna, Aug 10 2019: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(n+1) - x^k)^(n-k)  =  Sum_{n>=0} a(n)*x^(2*n).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(n+1) + x^k)^(n-k) * (-1)^k  =  Sum_{n>=0} a(n)*x^(2*n). (End)
a(n) = 2*A000005(2n) - 3*A000005(n). - Ridouane Oudra, Oct 15 2019
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = 2*log(2)-1. - Amiram Eldar, Mar 01 2023

New definition from Vladeta Jovovic, Jan 27 2002

A279060 Number of divisors of n of the form 6*k + 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1

Offset: 0

Ilya Gutkovskiy, Dec 05 2016

Möbius transform is the period-6 sequence {1, 0, 0, 0, 0, 0, ...}.

			a(14) = 2 because 14 has 4 divisors {1,2,7,14} among which 2 divisors {1,7} are of the form 6*k + 1.

Antti Karttunen, Table of n, a(n) for n = 0..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001227, A001817, A001826, A001876, A035218, A140213, A188169, A319995, A320001.

Cf. A001620, A016629, A222457 (psi(1/6)).

nmax = 120; CoefficientList[Series[Sum[x^k/(1 - x^(6 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 120; CoefficientList[Series[Sum[x^(6 k + 1)/(1 - x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
Table[Count[Divisors[n],?(Mod[#,6]==1&)],{n,0,120}] (* _Harvey P. Dale, Apr 27 2018 *)

A279060(n) = if(!n,n,sumdiv(n, d, (1==(d%6)))); \\ Antti Karttunen, Jul 09 2017

from sympy import divisors
def A279060(n): return sum(d%6 == 1 for d in divisors(n)) # David Radcliffe, Jun 19 2025

G.f.: Sum_{k>=1} x^k/(1 - x^(6*k)).
G.f.: Sum_{k>=0} x^(6*k+1)/(1 - x^(6*k+1)).
From Antti Karttunen, Oct 03 2018: (Start)
a(n) = A320001(n) + [1 == n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
a(n) = A035218(n) - A319995(n). (End)
a(n) = (A035218(n) + A035178(n)) / 2. - David Radcliffe, Jun 19 2025
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,6) - (1 - gamma)/6 = 0.686263..., gamma(1,6) = -(psi(1/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A319995 Number of divisors of n of the form 6*k + 5.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A035218, A279060, A320005.

Cf. A001620, A016629, A222458 (psi(5/6)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A319995(n) = if(!n,n,sumdiv(n, d, (5==(d%6))));

a(n) = A035218(n) - A279060(n).
G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(6*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,6) - (1 - gamma)/6 = -0.220635..., gamma(5,6) = -(psi(5/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A320015 Number of proper divisors of n that are either of the form 6k+1 or 6k + 5.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 1, 1, 3, 2, 2, 2, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 2, 4, 1, 1, 1, 2, 3, 2, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 2, 2, 1, 2, 3, 2, 2, 2, 3, 1, 1, 3, 2, 3, 1, 2, 1, 2, 4

Offset: 1

Antti Karttunen, Oct 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001620, A035218, A232991, A293895, A293896, A320001, A320005.

a[n_] := DivisorSum[n, 1 &, # < n && MemberQ[{1, 5}, Mod[#, 6]] &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A320015(n) = if(!n,n,sumdiv(n, d, (d

a(n) = A320001(n) + A320005(n).
a(n) = A035218(n) - ch15(n), where ch15 is the characteristic function of numbers of the form +-1 mod 6, i.e., ch15(n) = A232991(n-1).
Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = 2*(gamma + log(12)/4 - 1)/3 = 0.132294..., and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 3, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 3, 6, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 6, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Aug 26 2023

First differs from A034444 at n = 25.

The sum of these divisors is A365211(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A007310, A035218, A065330, A065331, A069733, A365211.

Cf. A001620, A013661, A073002.

f[p_, e_] := If[p <= 3 , 2, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] <= 3, 2, f[i,2]+1));}

Multiplicative with a(p^e) = 2 for p = 2 and 3, and a(p^e) = e+1 for a prime p >= 5.
a(n) <= A000005(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) >= A034444(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) = A000005(A065330(n)) * A034444(A065331(n)).
Dirichlet g.f.: (1-1/4^s) * (1-1/9^s) * zeta(s)^2.
Sum_{k=1..n} a(k) ~ (2*n/3) * (log(n) + 2*gamma - 1 + 2*log(2)/3 + log(3)/4), where gamma is Euler's constant (A001620).

A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 16 2025

The sum of these divisors is A385045(n), and the largest of them is A065330(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

The unitary analog of A035218.
Cf. A007310, A034444, A065330, A385045.
Cf. A001620, A013661, A306016.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).

f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x <= 3, 1, 2), factor(n)[, 1]));

Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.
a(n) = A034444(n)/A382488(n).
a(n) <= A034444(n), with equality if and only if n is 5-rough.
a(n) <= A035218(n).
Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * (1/((1+1/2^s)*(1+1/3^s))).
Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A099751 Number of ways to write n as differences of (-4)-gonal numbers. If pe(n):=1/2n((e-2)*n+(4-e)) is the n-th e-gonal number, then a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-4.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0, 2, 3, 2, 0, 2, 2, 2, 0, 2, 2, 3, 0, 1, 2, 2, 0, 2, 4, 2, 0, 4, 1, 2, 0, 2, 4, 2, 0, 2, 2, 2, 0, 2, 3, 3, 0, 2, 2, 2, 0, 4, 4, 2, 0, 2, 2, 2, 0, 2, 5, 4, 0, 2, 2, 2, 0, 2, 2, 2, 0, 3, 2, 4, 0, 2, 6, 1, 0, 2, 2, 4, 0, 2, 4, 2, 0, 4, 2, 2, 0, 4, 4, 2, 0, 2, 3, 2, 0, 2, 4, 4

Offset: 1

Volker Schmitt (clamsi(AT)gmx.net), Nov 10 2004

			G.f. = x + x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^8 + x^9 + 2*x^11 + x^12 + 2*x^13 + ...
a(5)=2 because there are two ways of differences: First pe(3)-pe(-2)=(-15)-(-20)=5 and second pe(1)-pe(2)=(1)-(-4)=5, for e=-4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001227 for e in {3, -2, 6}, A048272 for e in {0, 1, 4, 8} and A035218 for e=-1.

Cf. A001620, A027748, A124010, A256232, A035191.

a099751 n = product $ zipWith f (a027748_row n) (a124010_row n)
   where f 2 e = e - 1; f 3 e = 1; f _ e = e + 1
-- Reinhard Zumkeller, Mar 20 2015

res:=1; ifac:=op(ifactors(i))[2]; for pfac in ifac do; if pfac[1]=2 then res:=res*(pfac[2]-1); else if pfac[1]<>3 then res:=res*(pfac[2]+1); fi; fi; od; a(i):=res;

a[ n_] := If[ n < 1, 0, If[ Divisible[n, 4], -1, 1] Sum[ KroneckerSymbol[ -3, d] (-1)^Quotient[ d, 3], {d, Divisors@n}]]; (* Michael Somos, Mar 19 2015 *)

{a(n) = if( n<1, 0, if( n%2==0, (valuation(n, 2) -1) * a(n / 2^valuation(n, 2)), if( n%3==0, a(n / 3^valuation(n, 3)), numdiv(n)))) }; /* Michael Somos, Sep 20 2005 */

{a(n) = if( n<1, 0, (-1)^(n%4 == 0) * sumdiv( n, d, (-1)^(d\3) * kronecker( -3, d)))}; /* Michael Somos, Nov 16 2011 */

Multiplicative with a(2^e) = e-1 if e>0, a(3^e) = 1, a(p^e) = e+1 if p>3.
Moebius transform is period 12 sequence [ 1, -1, 0, 1, 1, 0, 1, 1, 0, -1, 1, 0, ...].
G.f.: Sum_{k>0} (x^k - x^(2*k) + x^(4*k) + x^(5*k) + x^(7*k) + x^(8*k) - x^(10*k) + x^(11*k)) / (1 - x^(12*k)). - Michael Somos, Sep 20 2005
a(3*n) = a(n). a(4*n + 2) = 0. - Michael Somos, Nov 16 2011
a(4*n) = A035191(n). - Michael Somos, Mar 19 2015
From Amiram Eldar, Nov 30 2022: (Start)
Dirichlet g.f.: zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s))*(1 - 1/3^s).
Sum_{k=1..n} a(k) ~ n*log(n)/3 + (2*gamma - 1 + log(3)/2)*n/3, where gamma is Euler's constant (A001620). (End)

A366642 Lexicographically earliest sequence of distinct primes such that the sequence of ratios (number of divisors of n that are coprime to these primes)/(number of divisors of n) has an asymptotic mean 1/2.

Original entry on oeis.org

2, 3, 5, 149, 10771, 16575407, 39516855101743, 60095055821549024117399447, 96668175211190122501174866643973679330023904660323

Offset: 1

Amiram Eldar, Oct 15 2023

nonn

The sequence of the number of divisors of n that are coprime to these primes is A366643.
Equivalently, a(n) is the lexicographically earliest sequence of distinct primes such that Product_{n>=1} (a(n)-1) * log(a(n)/(a(n)-1)) = 1/2.
The next term has 99 digits and is too large to be included in the data section.

			The asymptotic mean of (number of divisors of n that are coprime to 2)/A000005(n) = A001227(n)/A000005(n) is log(2) = 0.693... > 1/2. Therefore a(1) = 2.
The asymptotic mean of (number of divisors of n that are coprime to 2 and 3)/A000005(n) = A035218(n)/A000005(n) is 2*log(3/2)*log(2) = 0.562... > 1/2. Therefore a(2) = 3.
The asymptotic mean of (number of divisors of n that are coprime to 2, 3 and 5)/A000005(n) is 8*log(5/4)*log(3/2)*log(2) = 0.501... > 1/2. Therefore a(3) = 5.
The asymptotic mean of (number of divisors of n that are coprime to 2, 3, 5 and 7)/A000005(n) is 48*log(7/6)*log(5/4)*log(3/2)*log(2) = 0.464... < 1/2. Therefore a(4) is not 7.
The asymptotic mean of (number of divisors of n that are coprime to 2, 3, 5 and 149)/A000005(n) is 1184*log(149/148)*log(5/4)*log(3/2)*log(2) = 0.50002... > 1/2, and 149 is the least prime with this property. Therefore a(4) = 149.

Amiram Eldar, Table of n, a(n) for n = 1..13

Cf. A000005, A001227, A035218, A366643.

g[x_] := -(x-1)*Log[1-1/x]; seq[len_] := Module[{s = {}, r = 1/2, p = 1}, Do[p = NextPrime[InverseFunction[g][r]]; AppendTo[s, p]; r /= g[p], {len}]; s]; seq[7]

cp's OEIS Frontend

A048272 Number of odd divisors of n minus number of even divisors of n.

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A279060 Number of divisors of n of the form 6*k + 1.

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A319995 Number of divisors of n of the form 6*k + 5.

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A320015 Number of proper divisors of n that are either of the form 6*k+1 or 6*k + 5.

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A365210 The number of divisors d of n such that gcd(d, n/d) is a 5-rough number (A007310).

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A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

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A099751 Number of ways to write n as differences of (-4)-gonal numbers. If pe(n):=1/2*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-4.

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A320015 Number of proper divisors of n that are either of the form 6k+1 or 6k + 5.

A099751 Number of ways to write n as differences of (-4)-gonal numbers. If pe(n):=1/2n((e-2)*n+(4-e)) is the n-th e-gonal number, then a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-4.