A114003 -id:A114003 - cp's OEIS frontend

A152771 a(n) = sigma(n) - 2*d(n) + 1.

0, 0, 1, 2, 3, 5, 5, 8, 8, 11, 9, 17, 11, 17, 17, 22, 15, 28, 17, 31, 25, 29, 21, 45, 26, 35, 33, 45, 27, 57, 29, 52, 41, 47, 41, 74, 35, 53, 49, 75, 39, 81, 41, 73, 67, 65, 45, 105, 52, 82, 65, 87, 51, 105, 65, 105, 73, 83, 57, 145

Offset: 1

Omar E. Pol, Dec 14 2008

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A152770, A152772.

Table[DivisorSigma[1, n] - 2 DivisorSigma[0, n] + 1, {n, 60}] (* Ivan Neretin, Sep 30 2017 *)

a(n) = sigma(n) - 2*numdiv(n) + 1; \\ Michel Marcus, Sep 30 2017

a(n) = A000203(n) - 2*A000005(n) + 1 = A000203(n) - A114003(n) = A088580(n) - A062011(n). - Omar E. Pol, Sep 30 2017

G.f.: Sum_{k>=1} x^(3*k) / (1 - x^k)^2. - Ilya Gutkovskiy, Apr 24 2021

A114002 Expansion of g.f. x^k(1+x^(k+1))/(1-x^(k+1)).

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 0, 0, 0, 1, 2, 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, 2, 2, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Nov 12 2005

Inverse is A114004. Row sums are A114003.

			Triangle begins:
  1;
  2, 1;
  2, 0, 1;
  2, 2, 0, 1;
  2, 0, 0, 0, 1;
  2, 2, 2, 0, 0, 1;
  2, 0, 0, 0, 0, 0, 1;
  ...

Cf. A051731, A114003, A114004.

T[n_,k_]:=SeriesCoefficient[x^k(1+x^(k+1))/(1-x^(k+1)),{x,0,n}]; Table[T[n,k],{n,0,13},{k,0,n}] //Flatten (* Stefano Spezia, Sep 08 2023 *)

Column k has g.f. x^k(1+x^(k+1))/(1-x^(k+1)).

Equals 2*A051731 - I, I = Identity matrix. - Gary W. Adamson, Nov 07 2007

A360999 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

Original entry on oeis.org

2, 2, 3, 4, 3, 6, 3, 6, 5, 6, 3, 10, 3, 6, 7, 8, 3, 10, 3, 10, 7, 6, 3, 14, 5, 6, 7, 10, 3, 14, 3, 10, 7, 6, 7, 16, 3, 6, 7, 14, 3, 14, 3, 10, 11, 6, 3, 18, 5, 10, 7, 10, 3, 14, 7, 14, 7, 6, 3, 22, 3, 6, 11, 12, 7, 14, 3, 10, 7, 14, 3, 22, 3, 6, 11, 10, 7, 14

Offset: 1

Pontus von Brömssen, Feb 28 2023

nonn

Second column of A360998.
Essentially the same as A086369.
Cf. A000005, A000035, A114003, A360631, A361004.

a(n) = 2*A000005(n) - 1 - [n even] = A114003(n) + A000035(n) - 1 for n >= 2.

A128184 A051731 * A097806.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Feb 17 2007

Row sums = A114003: (1, 3, 3, 5, 3, 7, 3, 7, 5, 7, ...).

			First few rows of the triangle:
  1;
  2, 1;
  1, 1, 1;
  2, 1, 1, 1;
  1, 0, 0, 1, 1;
  2, 2, 1, 0, 1, 1;
  ...

Cf. A051731, A097806, A114003.

A051731 * A097806, (inverse Moebius transform of A097806).

A144515 Triangle read by rows: A051731 * A103451.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, Nov 21 2007

Row sums = A114003: (1, 3, 3, 5, 3, 7, 3, 7, 5, 7,...).

Left border = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4,...).

			First few rows of the triangle are:
1;
2, 1;
2, 0, 1;
3, 1, 0, 1;
2, 0, 0, 0, 1;
4, 1, 1, 0, 0, 1;
2, 0, 0, 0, 0, 0, 1;
...

Cf. A114003, A000005, A103451.

A051731 * A103451 as infinite lower triangular matrices. Left border of A051731 (all 1's) is replaced with A000005: (1, 2, 2, 3, 2, 4,...).

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 *)

A365540 Antidiagonal sums of A365539.

Original entry on oeis.org

0, 1, 5, 12, 24, 39, 61, 86, 118, 155, 199, 246, 304, 365, 433, 508, 592, 679, 777, 878, 990, 1109, 1235, 1364, 1508, 1657, 1813, 1976, 2150, 2327, 2519, 2714, 2920, 3133, 3353, 3580, 3824, 4071, 4325, 4586, 4862, 5141, 5435, 5732, 6040, 6359, 6685, 7014, 7362

Offset: 0

Stefano Spezia, Sep 08 2023

nonn

Cf. A000005, A114003 (2nd differences), A365539.

A365539[n_,k_]:=SeriesCoefficient[(1+x^k)/((1-x)^2*(1-x^k)),{x,0,n}]; Table[Sum[A365539[n-k,k], {k,n}], {n,0,48}] (* or *)
Table[Sum[Sum[2  DivisorSigma[0,k]-1,{k,m}],{m,n}],{n,0,48}]

a(n) = Sum_{m=1..n} (Sum_{k=1..m} (2*A000005(k) - 1)).

cp's OEIS Frontend

A152771 a(n) = sigma(n) - 2*d(n) + 1.

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A114002 Expansion of g.f. x^k(1+x^(k+1))/(1-x^(k+1)).

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A360999 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling).

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A128184 A051731 * A097806.

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A144515 Triangle read by rows: A051731 * A103451.

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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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A365540 Antidiagonal sums of A365539.

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