A279060 -id:A279060 - cp's OEIS frontend

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

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

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)

A320001 Number of proper divisors of n of the form 6*k + 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 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. A279060, A293895, A319991, A320003, A320005, A320015.

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

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

A320001(n) = if(!n,n,sumdiv(n, d, (d

a(n) = A279060(n) - [+1 = n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
a(n) = A320015(n) - A320005(n).
a(n) = A007814(A319991(n)).
G.f.: Sum_{k>=1} x^(12*k-10) / (1 - x^(6*k-5)). - 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(1,6) - (2 - gamma)/6 = 0.519597..., 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

A140213 Product_{h|n and h mod 6 = 1} h; product of divisors of n of the form 6*k + 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 19, 1, 7, 1, 1, 1, 25, 13, 1, 7, 1, 1, 31, 1, 1, 1, 7, 1, 37, 19, 13, 1, 1, 7, 43, 1, 1, 1, 1, 1, 343, 25, 1, 13, 1, 1, 55, 7, 19, 1, 1, 1, 61, 31, 7, 1, 13, 1, 67, 1, 1, 7, 1, 1, 73, 37, 25, 19, 7, 13, 79, 1, 1, 1, 1, 7, 85, 43, 1, 1, 1, 1, 8281

Offset: 1

R. J. Mathar, Jun 27 2008

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A170824, A248909, A279060.

A140213 := proc(n)
    a := 1;
    for d in numtheory[divisors](n) do
        if modp(d,6) = 1 then
            a := a*d ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Dec 15 2015

A140213(n) = { my(m=1); fordiv(n, d, if(1==(d%6), m *= d)); (m); }; \\ Antti Karttunen, Jul 09 2017

More terms from D. S. McNeil, Mar 24 2009

A359309 Number of divisors of 6n-5 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A279060, A359305, A359306, A359307, A359308.

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

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

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

a(n) = A279060(6*n-5).

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

A359306 Number of divisors of 6n-2 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A279060, A359305, A359307, A359308, A359309.

a[n_] := DivisorSum[6*n-2, 1 &, Mod[#, 6] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(6*n-2, d, d%6==1);
```

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

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

a(n) = A279060(6*n-2).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-2)).
G.f.: Sum_{k>0} x^(4*k-3)/(1 - x^(6*k-5)).

A359307 Number of divisors of 6n-3 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A279060, A359305, A359306, A359308, A359309.

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

PARI
```
a(n) = sumdiv(6*n-3, d, d%6==1);
```

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

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

a(n) = A279060(6*n-3).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-3)).
G.f.: Sum_{k>0} x^(3*k-2)/(1 - x^(6*k-5)).

A359308 Number of divisors of 6n-4 of form 6k+1.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Dec 25 2022

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

Cf. A279060, A359305, A359306, A359307, A359309.

a[n_] := DivisorSum[6*n-4, 1 &, Mod[#, 6] == 1 &]; Array[a, 100] (* Amiram Eldar, Aug 16 2023 *)

PARI
```
a(n) = sumdiv(6*n-4, d, d%6==1);
```

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

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

a(n) = A279060(6*n-4).
G.f.: Sum_{k>0} x^k/(1 - x^(6*k-4)).
G.f.: Sum_{k>0} x^(2*k-1)/(1 - x^(6*k-5)).

A304978 Numbers that can be expressed in more than one way as 6xy + x + y with x >= y > 0.

Original entry on oeis.org

106, 155, 197, 204, 253, 288, 302, 351, 379, 400, 421, 449, 470, 498, 504, 535, 547, 554, 561, 596, 645, 652, 687, 694, 704, 729, 743, 779, 782, 792, 820, 834, 841, 873, 890, 904, 925, 939, 953, 988, 1016, 1029, 1037, 1042, 1054, 1079, 1086, 1107, 1121, 1135, 1184, 1198, 1204, 1211, 1219, 1233, 1254, 1276, 1282, 1289, 1329

Offset: 1

Pedro Caceres, May 22 2018

nonn

Is it possible to find a closed form formula for this sequence?

Numbers k such that 6*k+1 has at least 5 divisors == 1 (mod 6). - Robert Israel, Jan 20 2019

			106 is in this sequence because 106 can be expressed in two different ways as 6xy + x + y: 6*8*2 + 8 + 2 and 6*15*1 + 15 + 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A067611. A279060.

filter:= proc(n) nops(select(t -> t mod 6 =1, numtheory:-divisors(6*n+1)))>= 5 end proc:
select(filter, [$1..2000]); # Robert Israel, Jan 20 2019

Select[Range[1329], 2 == Length@ FindInstance[ 6*x*y+x+y == # && x >= y > 0, {x, y}, Integers, 2] &] (* Giovanni Resta, May 29 2018 *)

is(n) = my(i=0); for(x=1, n, for(y=1, x, if(n==6*x*y+x+y, i++; if(i==2, return(1))))); 0 \\ Felix Fröhlich, May 29 2018

from sympy import divisors
def ok(n): return sum(d%6 == 1 for d in divisors(6*n+1)) > 4
print([n for n in range(1330) if ok(n)]) # David Radcliffe, Jun 19 2025

cp's OEIS Frontend

A319995 Number of divisors of n of the form 6*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

A320001 Number of proper divisors of n of the form 6*k + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A140213 Product_{h|n and h mod 6 = 1} h; product of divisors of n of the form 6*k + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Extensions

A359309 Number of divisors of 6*n-5 of form 6*k+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A359306 Number of divisors of 6*n-2 of form 6*k+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

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.

A359309 Number of divisors of 6n-5 of form 6k+1.

A359306 Number of divisors of 6n-2 of form 6k+1.

A359307 Number of divisors of 6n-3 of form 6k+1.

A359308 Number of divisors of 6n-4 of form 6k+1.