A320001 -id:A320001 - 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

A319991 a(n) = Product_{d|n, dA019565(d)^[1 == d mod 3].

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 10, 2, 2, 2, 10, 2, 60, 2, 10, 2, 2, 2, 210, 60, 2, 2, 10, 2, 140, 2, 300, 2, 42, 2, 110, 2, 2, 60, 10, 2, 132, 140, 210, 2, 60, 2, 1650, 2, 2, 2, 110, 60, 6468, 2, 700, 2, 2, 2, 115500, 132, 2, 2, 210, 2, 4620, 60, 110, 140, 330, 2, 390, 2, 1260, 2, 10, 2, 260, 308, 660, 60, 140, 2, 210210, 2, 2, 2, 115500, 2, 1092, 2

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A019565, A293214, A293895, A293897, A319990, A319992, A320001, A320011 (rgs-transform).

Cf. also A293221.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A319991(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

a(n) = Product_{d|n, dA019565(d)^[1 == d mod 3].
a(n) = A293214(n) / (A319990(n)*A319992(n)).
For all n >= 1:
A007814(a(n)) = A320001(n).
A048675(a(n)) = A293897(n).
A195017(a(n)) = A293895(n) mod 3.

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

A320011 Filter sequence combined from those proper divisors d of n for which +1 == d (mod 3); Restricted growth sequence transform of A319991.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 2, 5, 4, 2, 2, 3, 2, 6, 2, 7, 2, 8, 2, 9, 2, 2, 4, 3, 2, 10, 6, 5, 2, 4, 2, 11, 2, 2, 2, 9, 4, 12, 2, 13, 2, 2, 2, 14, 10, 2, 2, 5, 2, 15, 4, 9, 6, 16, 2, 17, 2, 18, 2, 3, 2, 19, 20, 21, 4, 6, 2, 22, 2, 2, 2, 14, 2, 23, 2, 11, 2, 8, 24, 25, 15, 2, 10, 9, 2, 26, 2, 27, 2, 28, 2, 29, 4

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

For all i, j:
  a(i) = a(j) => A320001(i) = A320001(j),
  a(i) = a(j) => A293897(i) = A293897(j).

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

Cf. A293223, A319991, A320010, A320012, A320013, A320014.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319991(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
v320011 = rgs_transform(vector(up_to,n,A319991(n)));
A320011(n) = v320011[n];

A320003 Number of proper divisors of n of the form 6*k + 3.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2018

Number of divisors of n that are odd multiples of 3 and less than n.

			For n = 18, of its five proper divisors [1, 2, 3, 6, 9] only 3 and 9 are odd multiples of three, thus a(18) = 2.
For n = 108, the odd part is 27 for which 27/3 has 3 divisors. As 108 is even, we don't subtract 1 from that 3 to get a(108) = 3. - _David A. Corneth_, 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. A319990, A320001, A320005.

Cf. A001620, A016629, A020759 (psi(1/2)).

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

A320003(n) = if(!n,n,sumdiv(n, d, (d

a(n) = if(n%3==0, my(v=valuation(n, 2)); n>>=v; numdiv(n/3)-(!v), 0) \\ David A. Corneth, Oct 03 2018

a(n) = Sum_{d|n, dA000035(d))*A079978(d).
a(n) = A007814(A319990(n)).
a(4*n) = a(2*n). - David A. Corneth, Oct 03 2018
G.f.: Sum_{k>=1} x^(12*k-6) / (1 - x^(6*k-3)). - 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(3,6) - (2 - gamma)/6 = -0.208505..., gamma(3,6) = -(psi(1/2) + 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

cp's OEIS Frontend

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

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A319991 a(n) = Product_{d|n, dA019565(d)^[1 == d mod 3].

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

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A320011 Filter sequence combined from those proper divisors d of n for which +1 == d (mod 3); Restricted growth sequence transform of A319991.

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A320003 Number of proper divisors of n of the form 6*k + 3.

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