A130773 -id:A130773 - cp's OEIS frontend

A144396 The odd numbers greater than 1.

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133

Offset: 1

Paul Curtz, Oct 03 2008

Last number of the n-th row of the triangle described in A142717.
If negated, these are also the values at local minima of the sequence A141620.
a(n) is the shortest leg of the n-th Pythagorean triple with consecutive longer leg and hypotenuse. The n-th such triple is given by (2n+1,2n^2+2n, 2n^2+2n+1), so that the longer legs are A046092(n) and the hypotenuses are A099776(n). - Ant King, Feb 10 2011
Numbers k such that the symmetric representation of sigma(k) has a pair of bars as its ends (cf. A237593). - Omar E. Pol, Sep 28 2018
Numbers k such that there is a prime knot with k crossings and braid index 2. (IS this true with "prime" removed?) - Charles R Greathouse IV, Feb 14 2023

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1)

Complement of A004275 and of A004277.
Cf. A000217, A002378, A005408, A007395, A046092, A099776, A237593, A254858.
Essentially the same as A140139, A130773, A062545, A020735, A005818.

List([3,5..200],n->n); # Muniru A Asiru, Sep 28 2018

a144396 = (+ 1) . (* 2)
a144396_list = [3, 5 ..] -- Reinhard Zumkeller, Feb 09 2015

seq(n,n=3..200,2); # Muniru A Asiru, Sep 28 2018

Range[3, 200, 2] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

a(n)=2*n+1 \\ Charles R Greathouse IV, Feb 14 2023

a(n) = A005408(n+1) = A000290(n+1) - A000290(n).
G.f.: x*(3-x)/(1-x)^2. - Jaume Oliver Lafont, Aug 30 2009
a(n) = A254858(n-1,2). - Reinhard Zumkeller, Feb 09 2015

Edited by R. J. Mathar, May 21 2009

A140139 Binomial transform of [1, 1, 2, -3, 4, -5, 6, -7, ...].

Original entry on oeis.org

1, 2, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141

Offset: 1

Gary W. Adamson, May 09 2008

Apart from initial term, identical to A130773 if offsets are ignored. - R. J. Mathar, May 11 2008

			a(4) = 7 = (1, 3, 3, 1) dot (1, 1, 2, -3) = (1 + 3 + 6 - 3).

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A007318, A130773.

LinearRecurrence[{2,-1},{1,2,5,7},71] (* Stefano Spezia, Apr 20 2025 *)

Equals A007318 * [1, 1, 2, -3, 4, -5, 6, -7, ...].
Sums of antidiagonal terms of the following array: 1, 1, 1, 1, 1, ... 1, 3, 5, 7, 9, ... 1, 1, 1, 1, 1, ...
O.g.f.: x*(1 + 2*x^2 - x^3)/(1 - x)^2. - R. J. Mathar, May 11 2008
E.g.f.: 1 - x^2/2 - exp(x)*(1 - 2*x). - Stefano Spezia, Apr 20 2025

A163985 Sum of all isolated parts of all partitions of n.

Original entry on oeis.org

0, 1, 2, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113

Offset: 0

Omar E. Pol, Aug 14 2009

Note that for n >= 3 the isolated parts of all partitions of n are n and n-1.

			For n=4, the five partitions of 4 are {(4);(2,2);(3,1);(2,1,1);(1,1,1,1)}. Since 1 and 2 are repeated parts and 3 and 4 are not repeated parts (or isolated parts) then a(4) = 3 + 4 = 7.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Omar E. Pol, Illustration of the shell model of partitions (2D view).
Omar E. Pol, Illustration of the shell model of partitions (3D view).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000041, A005408, A140139, A163986.

[0,1,2] cat [2*n-1: n in [3..60]]; // Vincenzo Librandi, Dec 23 2015

Join[{0, 1, 2}, Table[2 n - 1, {n, 3, 60}]] (* Vincenzo Librandi, Dec 23 2015 *)

a(n) = if (n<3, n, 2*n-1); \\ Michel Marcus, Dec 23 2015

a(n) = n for n<3, a(n) = 2*n-1 for n>=3.
a(n) = A140139(n), n>=1.
a(n) = A130773(n-1), n >=2. - R. J. Mathar, Jan 25 2023
From Stefano Spezia, Apr 21 2025: (Start)
G.f.: x*(1 + 2*x^2 - x^3)/(1 - x)^2.
E.g.f.: 1 - x^2/2 - exp(x)*(1 - 2*x). (End)

A223134 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n.

Original entry on oeis.org

1, 4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 0

Robert Price, Jun 12 2013

nonn

Appears to be essentially the same as A176271, A140139, A130773, A062545. - R. J. Mathar, Aug 23 2024

Robert Price, Table of n, a(n) for n = 0..100

Cf. A226357, A226359, A222945, A222947, A223133.

f[n_] := Length[Complement[Union[Flatten[Table[If[i*j*k <= n, {i + j + k}], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

A380099 a(n) is the n-digit numerator of the fraction h/k with h and k coprime positive integers at which abs((h/k)^4-Pi) is minimal.

Original entry on oeis.org

4, 97, 888, 9551, 13549, 505311, 4601995, 87956765, 298132602

Offset: 1

Stefano Spezia, Jan 12 2025

a(1)^4 = 4^4 = 256 corresponds to the numerator of A210621.

It appears that the number of correct decimal digits of Pi obtained from the fraction a(n)/A380100(n) is A130773(n-1) for n > 1 (see Spezia in Links). - Stefano Spezia, Apr 20 2025

			  n               (h/k)^4    approximated value
  -   -------------------    ------------------
  1               (4/3)^4    3.1604938271604...
  2             (97/73)^4    3.1174212867620...
  3           (888/667)^4    3.1415829223858...
  4         (9551/7174)^4    3.1415927852873...
  5       (13549/10177)^4    3.1415926560044...
  ...

Cf. A000796, A092040, A130773, A210621.

Cf. A355622, A364844, A380100 (denominator).

nmax = 3; a = {}; hmin = kmin = 0; For[n = 1, n <= nmax, n++, minim = Infinity; For[h = 10^(n-1), h <10^n, h++, For[k = 1, k < 10^n/Pi^(1/4), k++, If[(dist = Abs[h^4/k^4-Pi]) < minim && GCD[h,k]==1, minim = dist; hmin=h; kmin = k]]]; AppendTo[a, hmin]]; a

a(6)-a(9) from Kritsada Moomuang, Apr 17 2025

A380100 a(n) is the denominator of the fraction h/k with h and k coprime positive integers at which abs((h/k)^4-Pi) is minimal, with the numerator h of n digits.

Original entry on oeis.org

3, 73, 667, 7174, 10177, 379552, 3456676, 66066573, 223935013

Offset: 1

Stefano Spezia, Jan 12 2025

a(1)^4 = 3^4 = 81 corresponds to the denominator of A210621.

It appears that the number of correct decimal digits of Pi obtained from the fraction A380099(n)/a(n) is A130773(n-1) for n > 1 (see Spezia in Links). - Stefano Spezia, Apr 20 2025

			  n               (h/k)^4    approximated value
  -   -------------------    ------------------
  1               (4/3)^4    3.1604938271604...
  2             (97/73)^4    3.1174212867620...
  3           (888/667)^4    3.1415829223858...
  4         (9551/7174)^4    3.1415927852873...
  5       (13549/10177)^4    3.1415926560044...
  ...

Cf. A000583, A000796, A130773, A210621.

Cf. A355623, A364845, A380099 (numerator).

nmax = 3; a = {}; hmin = kmin = 0; For[n = 1, n <= nmax, n++, minim = Infinity; For[h = 10^(n-1), h <10^n, h++, For[k = 1, k < 10^n/Pi^(1/4), k++, If[(dist = Abs[h^4/k^4-Pi]) < minim && GCD[h,k]==1, minim = dist; hmin=h; kmin = k]]]; AppendTo[a, kmin]]; a

a(6)-a(9) from Kritsada Moomuang, Apr 17 2025

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A144396 The odd numbers greater than 1.

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A140139 Binomial transform of [1, 1, 2, -3, 4, -5, 6, -7, ...].

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A163985 Sum of all isolated parts of all partitions of n.

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A223134 Number of distinct sums i+j+k with i,j,k >= 0, i*j*k <= n.

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A380099 a(n) is the n-digit numerator of the fraction h/k with h and k coprime positive integers at which abs((h/k)^4-Pi) is minimal.

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A380100 a(n) is the denominator of the fraction h/k with h and k coprime positive integers at which abs((h/k)^4-Pi) is minimal, with the numerator h of n digits.

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A223134 Number of distinct sums i+j+k with i,j,k >= 0, ijk <= n.