A083040 -id:A083040 - cp's OEIS frontend

A083039 Number of divisors of n that are <= 3.

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

Offset: 1

Daniele A. Gewurz (gewurz(AT)mat.uniroma1.it) and Francesca Merola (merola(AT)mat.uniroma1.it), May 06 2003

Periodic of period 6. Parker vector of the wreath product of S_3 and S, the symmetric group of a countable set.

			The divisors of 6 are 1, 2, 3 and 6. Of those divisors, 1, 2 and 3 are <= 3. That's three divisors, therefore, a(6) = 3. - _David A. Corneth_, Sep 30 2017

Antti Karttunen, Table of n, a(n) for n = 1..12000
D. A. Gewurz and F. Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., 6 (2003), 03.1.6
M. D. Hirschhorn, J. A. Sellers, Enumeration of unigraphical partitions, JIS 11 (2008) 08.4.6
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (-1, 0, 1, 1).

Cf. A000005, A007310, A008588, A047228, A083040, A138553.

LinearRecurrence[{-1, 0, 1, 1},{1, 2, 2, 2},90] (* Ray Chandler, Aug 26 2015 *)

PARI
```
a(n)=[3,1,2,2,2,1][n%6+1];
```

G.f.: x/(1-x) + x^2/(1-x^2) + x^3/(1-x^3).
a(n) = a(n-6) = a(-n).
a(n) = 11/6 - (1/2)*(-1)^n - (1/3)*cos(2*Pi*n/3) - (1/3)*3^(1/2)*sin(2*Pi*n/3). - Richard Choulet, Dec 12 2008
a(n) = Sum_{k=1..1} cos(n*(k - 1)/1*2*Pi)/1 + Sum_{k=1..2} cos(n*(k - 1)/2*2*Pi)/2 + Sum_{k=1..3} cos(n*(k - 1)/3*2*Pi)/3. - Mats Granvik, Sep 09 2012
a(n) = log_2(gcd(n,2) + gcd(n,6)). - Gary Detlefs, Feb 15 2014
a(n) = Sum_{d|n, d<=3} 1. - Wesley Ivan Hurt, Oct 30 2023

A338648 Number of divisors of n which are greater than 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 3, 2, 2, 3, 5, 1, 2, 2, 5, 1, 5, 1, 3, 4, 2, 1, 6, 2, 4, 2, 3, 1, 5, 3, 5, 2, 2, 1, 8, 1, 2, 4, 4, 3, 5, 1, 3, 2, 6, 1, 8, 1, 2, 4, 3, 3, 5, 1, 7, 3, 2, 1, 8, 3, 2, 2, 5, 1, 9, 3, 3, 2, 2, 3, 8, 1, 4, 4, 6, 1, 5, 1, 5, 6, 2, 1, 8, 1, 6

Offset: 1

Ilya Gutkovskiy, Apr 22 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers.

Column k=5 of A135539.

Cf. A000005, A001620, A023645, A032741, A083040, A321014, A338649, A338650, A338651, A338652, A338653.

Table[DivisorSum[n, 1 &, # > 4 &], {n, 1, 110}]
nmax = 110; CoefficientList[Series[Sum[x^(5 k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Drop[#, 1] &
nmax = 110; CoefficientList[Series[-Log[Product[(1 - x^k)^(1/k), {k, 5, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Drop[#, 1] &

a(n) = sumdiv(n, d, d>4); \\ Michel Marcus, Apr 22 2021; corrected Jun 13 2022

my(N=100, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=5, N, x^k/(1-x^k)))) \\ Seiichi Manyama, Jan 07 2023

G.f.: Sum_{k>=1} x^(5*k) / (1 - x^k).
L.g.f.: -log( Product_{k>=5} (1 - x^k)^(1/k) ).
a(n) = A000005(n) - A083040(n).
G.f.: Sum_{k>=5} x^k/(1 - x^k). - Seiichi Manyama, Jan 07 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 37/12), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

a(1)-a(4) prepended by David A. Corneth, Jun 13 2022

A138553 Table read by rows: T(n,k) is the number of divisors of k that are <= n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 1, 3, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 5, 1, 2, 2, 3, 2, 4, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 1, 4, 1, 4, 2, 2, 1, 5

Offset: 1

Franklin T. Adams-Watters, Mar 24 2008

Suggested by a question from Eric Desbiaux.
The row lengths are the lengths before the pattern for n repeats.
Antidiagonal sums A070824. [From Eric Desbiaux, Dec 10 2009]

			The first few rows start:
1, [A000012]
1, 2, [A000034]
1, 2, 2, 2, 1, 3, [A083039]
1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, [A083040]

Row lengths A003418, row sums A025529, frequencies in rows A096180.

Cf. A243987

lista(nrows) = {for (n=1, nrows, for (k=1, lcm(vector(n, i, i)), print1(sumdiv(k, d, d <=n), ", ");); print(););} \\ Michel Marcus, Jun 19 2014

T(n,k) = sum_{i|k, i<=n} 1.

Definition corrected by Franklin T. Adams-Watters, Jun 19 2014

cp's OEIS Frontend

A083039 Number of divisors of n that are <= 3.

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A338648 Number of divisors of n which are greater than 4.

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A138553 Table read by rows: T(n,k) is the number of divisors of k that are <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions