A320003 -id:A320003 - cp's OEIS frontend

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

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

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

A319990 a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 6, 1, 1, 90, 1, 1, 6, 1, 1, 1260, 1, 1, 6, 1, 1, 3150, 1, 1, 84, 1, 1, 18900, 1, 1, 6, 1, 1, 1455300, 1, 1, 6, 1, 1, 9900, 1, 1, 17640, 1, 1, 242550, 1, 1, 6, 1, 1, 19209960, 1, 1, 6, 1, 1, 764032500, 1, 1, 9240, 1, 1, 2340, 1, 1, 6, 1, 1, 7283776500, 1, 1, 1260, 1, 1, 35100, 1, 1, 38808, 1, 1, 94594500, 1, 1, 6, 1, 1

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

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

Cf. A293214, A319991, A319992, A320003, A320010 (rgs-transform).

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319990(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].
a(n) = A293214(n) / (A319991(n)*A319992(n)).
For all n >= 1:
A007814(a(n)) = A320003(n).
A195017(a(n)) = 0 mod 3.

A320010 Filter sequence combined from those proper divisors of n that are multiples of 3; Restricted growth sequence transform of A319990.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

For all i, j: a(i) = a(j) => A320003(i) = A320003(j).

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

Cf. A319990, A320011, A320012, 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
A319990(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
v320010 = rgs_transform(vector(up_to,n,A319990(n)));
A320010(n) = v320010[n];

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A319990 a(n) = Product_{d|n, dA019565(d)^[0 == d mod 3].

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A320010 Filter sequence combined from those proper divisors of n that are multiples of 3; Restricted growth sequence transform of A319990.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI