A069930 -id:A069930 - cp's OEIS frontend

A023645 a(n) = tau(n)-1 if n is odd or tau(n)-2 if n is even.

0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 3, 1, 4, 1, 4, 3, 2, 1, 6, 2, 2, 3, 4, 1, 6, 1, 4, 3, 2, 3, 7, 1, 2, 3, 6, 1, 6, 1, 4, 5, 2, 1, 8, 2, 4, 3, 4, 1, 6, 3, 6, 3, 2, 1, 10, 1, 2, 5, 5, 3, 6, 1, 4, 3, 6, 1, 10, 1, 2, 5, 4, 3, 6, 1, 8, 4, 2, 1, 10, 3, 2, 3, 6, 1, 10, 3, 4, 3, 2, 3, 10, 1, 4, 5, 7, 1, 6, 1, 6

Offset: 1

N. J. A. Sloane

Vertex-transitive graphs of valency 2 with n nodes.
Number of values of k such that n+2 divided by k leaves a remainder 2. - Amarnath Murthy, Aug 01 2002
Number of divisors of n that are less than n/2. - Peter Munn, Mar 31 2017, or equivalently, number of divisors of n that are greater than 2. - Antti Karttunen, Feb 20 2023
For n > 2, a(n) is the number of planar arrangements of equal-sized regular n-gons such that their centers lie on a circle and neighboring n-gons have an edge in common. - Peter Munn, Apr 23 2017
Number of partitions of n into two distinct parts such that the smaller divides the larger. - Wesley Ivan Hurt, Dec 21 2017

			x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + 4*x^12 + ...

CRC Handbook of Combinatorial Designs, 1996, p. 649.

T. D. Noe, Table of n, a(n) for n = 1..10000
Felix Fröhlich et al., Rings of regular polygons, SeqFan thread, March 26 2017.
Gordon Royle, Transitive Graphs

Cf. A000005, A001620, A023637-A023647.
Cf. A014405, A049992, A069930, A095374, A296955, A321014.
Second column of A072528.

with(numtheory); f := n->if n mod 2 = 1 then tau(n)-1 else tau(n)-2; fi;

Table[s = DivisorSigma[0, n]; If[OddQ[n], s - 1, s - 2], {n, 100}] (* T. D. Noe, Nov 18 2013 *)
Array[DivisorSigma[0, #] - 1 - Boole@ EvenQ@ # &, 104] (* Michael De Vlieger, Apr 25 2017 *)

{a(n) = if( n<1, 0, numdiv(n) - 2 + n%2)} /* Michael Somos, Apr 29 2003 */

a(n) = sumdiv(n, d, d < n/2); \\ Michel Marcus, Apr 01 2017

G.f.: Sum_{k>0} x^(3*k) / (1 - x^k). - Michael Somos, Apr 29 2003.
a(2*n) = A069930(n). a(2*n + 1) = A095374(n). - Michael Somos, Aug 30 2012
a(n) = A072528(n+2,2) for n > 2. - Peter Munn, May 14 2017
From Peter Bala, Jan 13 2021: (Start)
a(n) = Sum_{ d|n, d < n/2 } 1. Cf. A296955.
G.f.: Sum_{k >= 3} x^k/(1 - x^k). (End)
a(n) = A049992(n) - A014405(n). - Antti Karttunen, Feb 20 2023
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 5/2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 08 2024

More terms from Vladeta Jovovic, Dec 03 2001

A066660 Number of divisors of 2n excluding 1.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 3, 4, 5, 5, 3, 7, 3, 5, 7, 5, 3, 8, 3, 7, 7, 5, 3, 9, 5, 5, 7, 7, 3, 11, 3, 6, 7, 5, 7, 11, 3, 5, 7, 9, 3, 11, 3, 7, 11, 5, 3, 11, 5, 8, 7, 7, 3, 11, 7, 9, 7, 5, 3, 15, 3, 5, 11, 7, 7, 11, 3, 7, 7, 11, 3, 14, 3, 5, 11, 7, 7, 11, 3, 11, 9, 5, 3, 15, 7, 5, 7, 9, 3, 17, 7, 7, 7, 5, 7

Offset: 1

Benoit Cloitre, Jan 11 2002

a(n) is the number of integers of the form (n+k)/(n-k) for k=0,1,2,...,n-1.
Inverse Moebius transform of A040001 (offset 1).
The number of partitions of 2n into exactly two parts (2n-i,i) such that i divides (2n-i). - Wesley Ivan Hurt, Dec 22 2013

			a(4)=3 because (4+0)/(4-0), (4+2)/(4-2), (4+3)/(4-3) are integers.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A000005, A040001, A234306.

[DivisorSigma(0,2*n) -1: n in [1..100]]; // G. C. Greubel, Feb 13 2019

with(numtheory); A066660:=n->tau(2*n)-1; seq(A066660(n), n=1..100); # Wesley Ivan Hurt, Dec 22 2013

Table[DivisorSigma[0, 2 n] - 1, {n, 100}] (* Wesley Ivan Hurt, Dec 22 2013 *)

PARI
```
a(n)=if(n<1,0,sumdiv(n,d,(d>1)+d%2))
```

{a(n)=if(n<1, 0, numdiv(2*n)-1)} /* Michael Somos, Sep 03 2006 */

[sigma(2*n,0) -1 for n in (1..100)] # G. C. Greubel, Feb 13 2019

a(n) = A069930(n) + 1.
If n is an odd prime, then a(n)=3.
Asymptotic formula: 1/n*Sum(i=1, n, a(i)) = C*log(n) + o(log(n)) with C=3/2. [corrected by Vaclav Kotesovec, Feb 13 2019]
Also lim_{n -> infinity} card(iG.f.: Sum_{n>0} x^n(1 - x^(3n))/((1 - x^n)(1 - x^(2n))).
a(n) = d(2n) - 1, where d(n) is the number of divisors of n (A000005). - Wesley Ivan Hurt, Dec 22 2013
a(n) = n - A234306(n). - Antti Karttunen, Dec 22 2013
a(n) = Sum_{i=1..n} floor(2*n/i) - floor((2*n-1)/i). - Wesley Ivan Hurt, Nov 15 2017
Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 5), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019

A086327 Number of factors over Q in the factorization of the Chebyshev polynomial of the second kind U_n(x).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 3, 4, 4, 2, 6, 2, 4, 6, 4, 2, 7, 2, 6, 6, 4, 2, 8, 4, 4, 6, 6, 2, 10, 2, 5, 6, 4, 6, 10, 2, 4, 6, 8, 2, 10, 2, 6, 10, 4, 2, 10, 4, 7, 6, 6, 2, 10, 6, 8, 6, 4, 2, 14, 2, 4, 10, 6, 6, 10, 2, 6, 6, 10, 2, 13, 2, 4, 10, 6, 6, 10, 2, 10, 8, 4, 2, 14, 6, 4, 6, 8, 2, 16, 6, 6, 6, 4, 6, 12, 2, 7, 10, 10, 2, 10, 2, 8, 14, 4

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 30 2003

nonn

Initial terms are consistent with A069930(n+1). - Andrew Howroyd, Jul 10 2018

a(n) = A069930(n+1) at least for the first 1515 terms. - Antti Karttunen, Sep 25 2018

Antti Karttunen, Table of n, a(n) for n = 1..1515
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.

Cf. A001227 (number of factors of Chebyshev polynomials of 1st kind).

Cf. A069930, A086375, A086389.

a(n)={vecsum(factor(polchebyshev(n, 2, x))[, 2])} \\ Andrew Howroyd, Jul 10 2018

a(11) corrected and a(19)-a(85) from Andrew Howroyd, Jul 10 2018

Terms a(86)-a(105) from Antti Karttunen, Sep 25 2018

A348554 Irregular triangle read by rows: row n gives the divisors d of 2n with 1 < d < 2n, for n >= 2.

Original entry on oeis.org

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

Offset: 2

Wolfdieter Lang, Oct 22 2021

This gives the rows 2*n of A137510, for n >= 2.
The length of row n is A069930(n) = tau(2*n) - 2 = A099777(n) - 2.
The sum of row n is A346880(n) = A062731(n) - (2*n + 1).

			The irregular triangle T(n, k) begins:
n, 2*n / k 1  2  3  4  5  6  7 ...
----------------------------------
2,   4:    2
3,   6:    2  3
4,   8:    2  4
5,  10:    2  5
6   12:    2  3  4 6
7,  14:    2  7
8,  16:    2  4  8
9,  18:    2  3  6  9
10, 20:    2  4  5 10
11, 22:    2 11
12, 24:    2  3  4  6  8 12
13, 26:    2 13
14, 28:    2  4  7 14
15, 30:    2  3  5  6 10 15
16, 32:    2  4  8  1
17, 34:    2 17
18, 36:    2  3  4  6  9 12 18
19, 38:    2 19
20, 40:    2  4  5  8 10 20
...

Cf. A069930, A099777, A137510, A346880.

Flatten@Table[Select[Divisors[2n],1<#<2n&],{n,2,25}] (* Giorgos Kalogeropoulos, Oct 22 2021 *)

row(n) = select(x->((x>1) && (x<2*n)), divisors(2*n)); \\ Michel Marcus, Oct 23 2021

T(n, k) = A137510(2*n, k), for n >= 2 and k = 1, 2, ..., A069930(n).

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cp's OEIS Frontend

A023645 a(n) = tau(n)-1 if n is odd or tau(n)-2 if n is even.

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A066660 Number of divisors of 2n excluding 1.

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A086327 Number of factors over Q in the factorization of the Chebyshev polynomial of the second kind U_n(x).

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PARI

Extensions

A348554 Irregular triangle read by rows: row n gives the divisors d of 2*n with 1 < d < 2*n, for n >= 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A348554 Irregular triangle read by rows: row n gives the divisors d of 2n with 1 < d < 2n, for n >= 2.