A114002 -id:A114002 - cp's OEIS frontend

A114003 Rows sums of triangle A114002.

1, 3, 3, 5, 3, 7, 3, 7, 5, 7, 3, 11, 3, 7, 7, 9, 3, 11, 3, 11, 7, 7, 3, 15, 5, 7, 7, 11, 3, 15, 3, 11, 7, 7, 7, 17, 3, 7, 7, 15, 3, 15, 3, 11, 11, 7, 3, 19, 5, 11, 7, 11, 3, 15, 7, 15, 7, 7, 3, 23, 3, 7, 11, 13, 7, 15, 3, 11, 7, 15, 3, 23, 3, 7, 11, 11, 7, 15, 3, 19, 9, 7, 3, 23, 7, 7, 7, 15, 3, 23, 7, 11, 7, 7, 7, 23, 3

Offset: 1

Paul Barry, Nov 12 2005

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Also row sums of triangle A144515. - Gary W. Adamson, Nov 21 2007

Cf. A000005, A051731, A114002.

Table[2 DivisorSigma[0,n]-1,{n,97}] (* Stefano Spezia, Sep 08 2023 *)

N=66;x='x+O('x^N); /* that many terms */
Vec(sum(n=1,N,x^n*(1+x^n)/(1-x^n))) /* show terms */ /* Joerg Arndt, May 25 2011 */

A114003(n) = (2*numdiv(n))-1; \\ After Jovovic's formula. Antti Karttunen, May 25 2017

a(p) = 3, for primes p.
a(n) = 2*A000005(n) - 1. - Vladeta Jovovic, Sep 13 2006
Equals A051731 * [1, 2, 2, 2, ...]. - Gary W. Adamson, Sep 21 2007
G.f.: Sum_{n>0} x^n*(1+x^n)/(1-x^n). - Franklin T. Adams-Watters, Oct 09 2009

A114004 Inverse of triangle A114002.

Original entry on oeis.org

1, -2, 1, -2, 0, 1, 2, -2, 0, 1, -2, 0, 0, 0, 1, 6, -2, -2, 0, 0, 1, -2, 0, 0, 0, 0, 0, 1, -2, 2, 0, -2, 0, 0, 0, 1, 2, 0, -2, 0, 0, 0, 0, 0, 1, 6, -2, 0, 0, -2, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -10, 6, 2, -2, 0, -2, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, -2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, 6, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Nov 12 2005

Row sums are A114006. First column is A114005.

			Triangle begins
1;
-2, 1;
-2, 0, 1;
2,-2, 0, 1;
-2, 0, 0, 0, 1;
6,-2,-2, 0, 0, 1;
-2, 0, 0, 0, 0, 0, 1;
-2, 2, 0,-2, 0, 0, 0, 1;
2, 0,-2, 0, 0, 0, 0, 0, 1;
6,-2, 0, 0,-2, 0, 0, 0, 0, 1;

A235671 Triangle read by rows in which row n lists the proper divisors of n in increasing order, 2n, and the proper divisors of n in decreasing order.

Original entry on oeis.org

2, 1, 4, 1, 1, 6, 1, 1, 2, 8, 2, 1, 1, 10, 1, 1, 2, 3, 12, 3, 2, 1, 1, 14, 1, 1, 2, 4, 16, 4, 2, 1, 1, 3, 18, 3, 1, 1, 2, 5, 20, 5, 2, 1, 1, 22, 1, 1, 2, 3, 4, 6, 24, 6, 4, 3, 2, 1, 1, 26, 1, 1, 2, 7, 28, 7, 2, 1, 1, 3, 5, 30, 5, 3, 1, 1, 2, 4, 8, 32, 8, 4, 2, 1

Offset: 1

Omar E. Pol, Jan 24 2014

Numerators of a sequence related to the symmetric structure of sigma, which arises from the structure of A237593. The structure in the first two octants is transformed in a structure in the 6th and 7th octants, which is similar to an isosceles triangle.
Denominators are in A007395.
Row sums give A074400.
Row lengths is A114003 (see the Jovovic's formula in A114003).

			The irregular triangle begins:
2;
1, 4, 1;
1, 6, 1;
1, 2, 8, 2, 1;
1, 10, 1;
1, 2, 3, 12, 3, 2, 1;
1, 14, 1;
1, 2, 4, 16, 4, 2, 1;
1, 3, 18, 3, 1;
1, 2, 5, 20, 5, 2, 1;
1, 22, 1;
1, 2, 3, 4, 6, 24, 6, 4, 3, 2, 1;
...
Also:
1;
1/2, 2, 1/2;
1/2, 3, 1/2;
1/2, 1, 4, 1, 1/2;
1/2, 5, 1/2;
1/2, 1, 3/2, 6, 3/2, 1, 1/2;
1/2, 7, 1/2;
1/2, 1, 2, 8, 2, 1, 1/2;
1/2, 3/2, 9, 3/2, 1/2;
1/2, 1, 5/2, 10, 5/2, 1, 1/2;
1/2, 11, 1/2;
1/2, 1, 3/2, 2, 3, 12, 3, 2, 3/2, 1, 1/2;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A001065, A027750, A056538, A074400, A000203, A114002, A114003, A236104, A237591, A237593, A237270, A233772, A233773.

pd[n_]:=Module[{d=Most[Divisors[n]]},Flatten[Join[{d,{2n},Reverse[d]}]]]; Flatten[Array[pd,20]] (* Harvey P. Dale, Dec 22 2014 *)

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A114003 Rows sums of triangle A114002.

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A114004 Inverse of triangle A114002.

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A235671 Triangle read by rows in which row n lists the proper divisors of n in increasing order, 2n, and the proper divisors of n in decreasing order.

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