A319995 -id:A319995 - cp's OEIS frontend

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

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

A363807 Number of divisors of n of the form 7*k + 5.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Jun 23 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
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. A279061, A363795, A363805, A363806, A363808.
Cf. A284446, A319995.
Cf. A001620, A016630, A354631 (psi(5/7)).

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 5 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI
```
a(n) = sumdiv(n, d, d%7==5);
```

G.f.: Sum_{k>0} x^(5*k)/(1 - x^(7*k)).
G.f.: Sum_{k>0} x^(7*k-2)/(1 - x^(7*k-2)).
Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,7) - (1 - gamma)/7 = -0.169787..., gamma(5,7) = -(psi(5/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A035218 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 36.

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

Offset: 1

N. J. A. Sloane

a(n) is the number of factors (over Q) of the polynomial x^(2n) - x^n + 1. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003
This sequence is multiplicative. Just as (A001227)(n) is the number of ways to write n as differences of 3-gonal numbers, this sequence is the number of ways to write n as difference of (-1)-gonal numbers. If p_e(n):=1/2*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then 2*a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-1. - Volker Schmitt (clamsi(AT)gmx.net), Oct 11 2004
a(n) is the number of divisors of n not divisible by 2 or 3. For example, a(36)=1 because 1 is the only such divisor of 36. a(10) = 2 because we count the divisors 1 and 5. - Geoffrey Critzer, Feb 15 2015

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A035191, A000005, A001227, A279060, A319995, A320015, A001620.

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

nn = 81; f[list_, i] := list[[i]]; a = Prepend[Drop[Table[Boole[Min[FactorInteger[n][[All, 1]]] > 3], {n, 1, nn}], 1], 1]; b = Table[1, {nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 15 2015 *)
f[p_, e_] := If[p >= 5, e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

m=36; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, (d % 2) && (d % 3)); \\ Michel Marcus, Feb 16 2015

a(n) = d(6n) - d(3n) - d(2n) + d(n) where d() is the divisor function. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003
Multiplicative with a(2^e)=1, a(3^e)=1, a(p^e)=e+1 if p>3. Inverse Möbius transform is periodic with 1, 0, 0, 0, 1, 0. - Volker Schmitt (clamsi(AT)gmx.net), Oct 11 2004
Dirichlet g.f.: zeta(s)^2*(1 - 1/2^s)*(1 - 1/3^s). - Geoffrey Critzer, Feb 15 2015
From Antti Karttunen, Oct 03 2018: (Start)
a(n) = A279060(n) + A319995(n).
a(n) = A320015(n) + ch15(n), where ch15 is the characteristic function of numbers of the form +-1 mod 6, i.e., ch15(n) = A232991(n-1).
(End)
Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma + log(12)/2 - 1)/3, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 29 2019

More terms from Antti Karttunen, Oct 03 2018

A359305 Number of divisors of 6n-1 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

Also number of divisors of 6*n-1 of form 6*k+5.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A279060, A319995.
Cf. A000005, A001620, A016969, A078703, A359211, A359233.
Cf. A359306, A359307, A359308, A359309.
Cf. A359324, A359325, A359326, A359327.

a[n_] := DivisorSigma[0, 6*n - 1]/2; Array[a, 100] (* Amiram Eldar, Dec 26 2022 *)

PARI
```
a(n) = sumdiv(6*n-1, d, d%6==1);
```
PARI
```
a(n) = sumdiv(6*n-1, d, d%6==5);
```

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-x^(6*k-1))))

my(N=100, x='x+O('x^N)); Vec(sum(k=1, N, x^(5*k-4)/(1-x^(6*k-5))))

a(n) = A279060(6*n-1) = A319995(6*n-1).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-1)).
G.f.: Sum_{k>0} x^(5*k-4)/(1 - x^(6*k-5)).
From Amiram Eldar, Dec 26 2022: (Start)
a(n) = A000005(A016969(n-1))/2.
Sum_{k=1..n} a(k) = (log(n) + 2*gamma - 1 + 3*log(2) + 2*log(3))*n/6 + O(n^(1/3)*log(n)), where gamma is Euler's constant (A001620). (End)

A320005 Number of proper divisors of n of the form 6*k + 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 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. A293896, A319992, A319995, A320001, A320003, A320015.

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

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

A320005(n) = if(!n,n,sumdiv(n, d, (d

a(n) = A319995(n) - [+5 = n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = -1 mod 6, and 0 otherwise.
a(n) = A320015(n) - A320001(n).
a(n) = A007814(A319992(n)).
G.f.: Sum_{k>=1} x^(12*k-2) / (1 - x^(6*k-1)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,6) - (2 - gamma)/6 = -0.387302..., 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

A359327 Number of divisors of 6n-5 of form 6k+5.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A319995, A359305, A359324, A359325, A359326.

Cf. A359239, A359240, A359241.

a[n_] := DivisorSum[6*n-5, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Aug 14 2023 *)

PARI
```
a(n) = sumdiv(6*n-5, d, d%6==5);
```

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/(1-x^(6*k-1)))))

a(n) = A319995(6*n-5).

G.f.: Sum_{k>0} x^(5*k)/(1 - x^(6*k-1)).

A359324 Number of divisors of 6n-2 of form 6k+5.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A319995, A359305, A359325, A359326, A359327.

Cf. A359239, A359269, A359290.

a[n_] := DivisorSum[6*n-2, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Aug 14 2023 *)

PARI
```
a(n) = sumdiv(6*n-2, d, d%6==5);
```

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^(6*k-1)))))

my(N=100, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(5*k-3)/(1-x^(6*k-4)))))

a(n) = A319995(6*n-2).
G.f.: Sum_{k>0} x^(2*k)/(1 - x^(6*k-1)).
G.f.: Sum_{k>0} x^(5*k-3)/(1 - x^(6*k-4)).

A359325 Number of divisors of 6n-3 of form 6k+5.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A319995, A359305, A359324, A359326, A359327.

a[n_] := DivisorSum[6*n-3, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(6*n-3, d, d%6==5);
```

my(N=100, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^(6*k-1)))))

my(N=100, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(5*k-2)/(1-x^(6*k-3)))))

a(n) = A319995(6*n-3).
G.f.: Sum_{k>0} x^(3*k)/(1 - x^(6*k-1)).
G.f.: Sum_{k>0} x^(5*k-2)/(1 - x^(6*k-3)).

A359326 Number of divisors of 6n-4 of form 6k+5.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A319995, A359305, A359324, A359325, A359327.

a[n_] := DivisorSum[6*n-4, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Aug 14 2023 *)

PARI
```
a(n) = sumdiv(6*n-4, d, d%6==5);
```

my(N=100, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^(6*k-1)))))

my(N=100, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(5*k-1)/(1-x^(6*k-2)))))

a(n) = A319995(6*n-4).
G.f.: Sum_{k>0} x^(4*k)/(1 - x^(6*k-1)).
G.f.: Sum_{k>0} x^(5*k-1)/(1 - x^(6*k-2)).

A364048 Expansion of Sum_{k>0} x^(5k) / (1 + x^(6k)).

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

Offset: 1

Seiichi Manyama, Jul 03 2023

sign

Cf. A319995.

a[n_] := DivisorSum[n, (-1)^((#-5)/6) &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Jul 03 2023 *)

a(n) = sumdiv(n, d, (d%6==5)*(-1)^((d-5)/6));

G.f.: Sum_{k>0} (-1)^(k-1) * x^(6*k-1) / (1 - x^(6*k-1)).

a(n) = Sum_{d|n, d==5 (mod 6)} (-1)^((d-5)/6).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A363807 Number of divisors of n of the form 7*k + 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A035218 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 36.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A359305 Number of divisors of 6*n-1 of form 6*k+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

A320005 Number of proper divisors of n of the form 6*k + 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A359327 Number of divisors of 6*n-5 of form 6*k+5.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A359324 Number of divisors of 6*n-2 of form 6*k+5.

Views

Author

Keywords

Links

Crossrefs

Programs

A035218 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 36.

A359305 Number of divisors of 6n-1 of form 6k+1.

A359327 Number of divisors of 6n-5 of form 6k+5.

A359324 Number of divisors of 6n-2 of form 6k+5.

A359325 Number of divisors of 6n-3 of form 6k+5.

A359326 Number of divisors of 6n-4 of form 6k+5.

A364048 Expansion of Sum_{k>0} x^(5k) / (1 + x^(6k)).