A339443 -id:A339443 - cp's OEIS frontend

A340205 Number of primes in A339443 among the values of A339443(k) for k = 1..n.

0, 0, 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 7, 8, 9, 10, 10, 10, 11, 12, 12, 13, 14, 14, 14, 15, 16, 17, 17, 17, 17, 17, 18, 19, 19, 20, 21, 21, 21, 21, 21, 22, 23, 24, 24, 24, 25, 26, 26, 26, 26, 27, 27, 28, 29, 29, 29, 30, 31, 31, 31, 32, 32, 33, 33, 33, 34, 35, 35, 35, 35, 35

Offset: 1

Wesley Ivan Hurt, Dec 31 2020

nonn

			a(12) = 6; The first 12 values of A339443 are 1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 3, 2 (of which, 6 are prime).

Index entries for sequences related to partitions

Cf. A000720, A339443, A340203.

a(n) = Sum_{k=1..n} pi(c(k)) - pi(c(k)-1), where c = A339443.

A350326 Binomial transform of A339443(n).

Original entry on oeis.org

1, 2, 5, 11, 24, 52, 110, 227, 463, 947, 1956, 4073, 8501, 17695, 36654, 75585, 155396, 318958, 654018, 1339502, 2738706, 5586721, 11368212, 23081884, 46793949, 94805057, 192116284, 389627700, 791036691, 1607529164, 3268715492, 6647212980, 13512728367, 27449702179

Offset: 0

Wesley Ivan Hurt, Dec 24 2021

nonn

Index entries for sequences related to partitions

Cf. A339443.

a[n_] := (1 - (-1)^n) (1 + Floor[Sqrt[2 n - 1]])/2 - (((-1)^n - 2 n - 1)/2 + 2 Sum[Floor[(k + 1)/2], {k, -1 + Floor[Sqrt[2 n - 2 - (-1)^n]]}]) (-1)^n/2; Table[Sum[Binomial[n, i]*a[n - i + 1], {i, 0, n}], {n, 0, 40}]

a(n) = Sum_{k=0..n} binomial(n,k) * A339443(n-k+1).

A339399 Pairwise listing of the partitions of k into two parts (s,t), with 0 < s <= t ordered by increasing values of s and where k = 2,3,... .

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 2, 3, 1, 5, 2, 4, 3, 3, 1, 6, 2, 5, 3, 4, 1, 7, 2, 6, 3, 5, 4, 4, 1, 8, 2, 7, 3, 6, 4, 5, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 1, 11, 2, 10, 3, 9, 4, 8, 5, 7, 6, 6, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 1, 13, 2, 12, 3, 11

Offset: 1

Wesley Ivan Hurt, Dec 02 2020

a(n-1) and a(n) are the lesser and greater of a twin prime pair if and only if a(n) = a(n-1) + 2 where a(n-1) and a(n) are prime.

			                                                                     [1,9]
                                                     [1,7]   [1,8]   [2,8]
                                     [1,5]   [1,6]   [2,6]   [2,7]   [3,7]
                     [1,3]   [1,4]   [2,4]   [2,5]   [3,5]   [3,6]   [4,6]
     [1,1]   [1,2]   [2,2]   [2,3]   [3,3]   [3,4]   [4,4]   [4,5]   [5,5]
   k   2       3       4       5       6       7       8       9      10
  --------------------------------------------------------------------------
   k   Nondecreasing partitions of k
  --------------------------------------------------------------------------
   2   1,1
   3   1,2
   4   1,3,2,2
   5   1,4,2,3
   6   1,5,2,4,3,3
   7   1,6,2,5,3,4
   8   1,7,2,6,3,5,4,4
   9   1,8,2,7,3,6,4,5
  10   1,9,2,8,3,7,4,6,5,5
  ...

Index entries for sequences related to partitions

Cf. A103889, A339443.

Bisections: A122197 (odd), A199474 (even).

t[n_] := Flatten[Reverse /@ IntegerPartitions[n, {2}]]; Array[t, 14, 2] // Flatten (* Amiram Eldar, Dec 03 2020 *)
Table[(1 + (-1)^n) (1 + Floor[Sqrt[2 n - 1 - (-1)^n]])/2 - ((2 n + 1 - (-1)^n)/2 - 2 Sum[Floor[(k + 1)/2], {k, -1 + Floor[Sqrt[2 n - 2 - (-1)^n]]}]) (-1)^n/2, {n, 100}] (* Wesley Ivan Hurt, Dec 04 2020 *)

row(n) = vector(n\2, i, [i, n-i]);
tabf(nn) = for (n=2, nn, print(row(n))); \\ Michel Marcus, Dec 03 2020

a(n) = (1+(-1)^n)*(1+floor(sqrt(2*n-1-(-1)^n)))/2-((2*n+1-(-1)^n)/2-2 *Sum_{k=1..floor(sqrt(2*n-2-(-1)^n)-1)} floor((k+1)/2))*(-1)^n/2.

a(n) = A339443(A103889(n)). - Wesley Ivan Hurt, May 09 2021

A342769 Pairwise listing of the partitions of 2k into two parts, (s,t), with 0 < s <= t ordered by increasing values of s and where k = 1,2,... .

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 1, 5, 2, 4, 3, 3, 1, 7, 2, 6, 3, 5, 4, 4, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 1, 11, 2, 10, 3, 9, 4, 8, 5, 7, 6, 6, 1, 13, 2, 12, 3, 11, 4, 10, 5, 9, 6, 8, 7, 7, 1, 15, 2, 14, 3, 13, 4, 12, 5, 11, 6, 10, 7, 9, 8, 8, 1, 17, 2, 16, 3, 15, 4, 14, 5, 13, 6, 12, 7

Offset: 1

Wesley Ivan Hurt, Mar 21 2021

			                                                        [1,13]
                                               [1,11]   [2,12]
                                       [1,9]   [2,10]   [3,11]
                               [1,7]   [2,8]   [3, 9]   [4,10]
                       [1,5]   [2,6]   [3,7]   [4, 8]   [5, 9]
               [1,3]   [2,4]   [3,5]   [4,6]   [5, 7]   [6, 8]
       [1,1]   [2,2]   [3,3]   [4,4]   [5,5]   [6, 6]   [7, 7]
   2k    2       4       6       8       10      12       14
  --------------------------------------------------------------------------
   2k   Nondecreasing partitions of 2k
  --------------------------------------------------------------------------
   2   1,1
   4   1,3,2,2
   6   1,5,2,4,3,3
   8   1,7,2,6,3,5,4,4
  10   1,9,2,8,3,7,4,6,5,5
  12   1,11,2,10,3,9,4,8,5,7,6,6
  14   1,13,2,12,3,11,4,10,5,9,6,8,7,7
  ...

Index entries for sequences related to partitions

Cf. A000194, A052928, A103889, A339399, A339443, A342913.

a(n) = k + (k^2 + k - m)*(-1)^n / 2, where k = round(sqrt(m)) and m = 2*floor((n+1)/2).

a(n) = A342913(A103889(n)). - Wesley Ivan Hurt, May 09 2021

A342913 Pairwise listing of the partitions of 2k into two parts, (s,t), with 0 < t <= s ordered by decreasing values of s and where k = 1,2,... .

Original entry on oeis.org

1, 1, 3, 1, 2, 2, 5, 1, 4, 2, 3, 3, 7, 1, 6, 2, 5, 3, 4, 4, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 11, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6, 13, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 7, 15, 1, 14, 2, 13, 3, 12, 4, 11, 5, 10, 6, 9, 7, 8, 8, 17, 1, 16, 2, 15, 3, 14, 4, 13, 5, 12, 6, 11, 7

Offset: 1

Wesley Ivan Hurt, Mar 28 2021

nonn

			                                                        [13,1]
                                               [11,1]   [12,2]
                                       [9,1]   [10,2]   [11,3]
                               [7,1]   [8,2]   [9, 3]   [10,4]
                       [5,1]   [6,2]   [7,3]   [8, 4]   [9, 5]
               [3,1]   [4,2]   [5,3]   [6,4]   [7, 5]   [8, 6]
       [1,1]   [2,2]   [3,3]   [4,4]   [5,5]   [6, 6]   [7, 7]
   2k    2       4       6       8       10      12       14
  --------------------------------------------------------------------------
   2k   Decreasing partitions of 2k
  --------------------------------------------------------------------------
   2   1,1
   4   3,1,2,2
   6   5,1,4,2,3,3
   8   7,1,6,2,5,3,4,4
  10   9,1,8,2,7,3,6,4,5,5
  12   11,1,10,2,9,3,8,4,7,5,6,6
  14   13,1,12,2,11,3,10,4,9,5,8,6,7,7
  ...

Index entries for sequences related to partitions

Cf. A000194, A052928, A103889, A339399, A339443, A342769.

a(n) = k - (k^2 + k - m)*(-1)^n / 2, where k = round(sqrt(m)) and m = 2*floor((n+1-(-1)^n)/2).

a(n) = A342769(A103889(n)).

A102515 a(n) = floor(1 + sqrt(2n + 1)).

Original entry on oeis.org

2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 0

Giovanni Teofilatto, Mar 16 2005

The first occurrence of k appears at floor((k-1)^2/2), beginning with k=3. - Robert G. Wilson v, Mar 01 2015

If the sign inside the sqrt() is changed from "+" to "-", then the offset must be changed from 0 to 1 in order for the terms in the data to remain the same. - Robert G. Wilson v, Mar 01 2015

Table[Floor[1+Sqrt[2n+1]],{n,0,100}] (* Harvey P. Dale, Feb 28 2015 *)

With offset 1, a(n) = A339399(n) + A339443(n). - Wesley Ivan Hurt, Dec 31 2020

Definition corrected by Olivier Gérard & Harvey P. Dale, Feb 28 2015

Offset corrected by Robert G. Wilson v, Mar 01 2015

A349523 a(n) = Sum_{k=1..n} A339399(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 13, 14, 18, 20, 23, 24, 29, 31, 35, 38, 41, 42, 48, 50, 55, 58, 62, 63, 70, 72, 78, 81, 86, 90, 94, 95, 103, 105, 112, 115, 121, 125, 130, 131, 140, 142, 150, 153, 160, 164, 170, 175, 180, 181, 191, 193, 202, 205, 213, 217, 224, 229, 235, 236, 247, 249

Offset: 1

Wesley Ivan Hurt, Nov 20 2021

nonn

Partial sums of A339399.

Index entries for sequences related to partitions

Cf. A103889, A339399, A339443.

Table[Sum[((1 + (-1)^k) (1 + Floor[Sqrt[2 k - 1 - (-1)^k]])/2 - ((2 k + 1 - (-1)^k)/2 - 2 Sum[Floor[(i + 1)/2], {i, -1 + Floor[Sqrt[2 k - 2 - (-1)^k]]}]) (-1)^k/2), {k, n}], {n, 100}]

a(n) = Sum_{i=1..n} ((1+(-1)^i)*(1+floor(sqrt(2*i-1-(-1)^i)))/2-((2*i+1-(-1)^i)/2-2 *Sum_{k=1..floor(sqrt(2*i-2-(-1)^i)-1)} floor((k+1)/2))*(-1)^i/2).

a(n) = Sum_{k=1..n} A339443(A103889(k)).

cp's OEIS Frontend

A340205 Number of primes in A339443 among the values of A339443(k) for k = 1..n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A350326 Binomial transform of A339443(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A339399 Pairwise listing of the partitions of k into two parts (s,t), with 0 < s <= t ordered by increasing values of s and where k = 2,3,... .

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A342769 Pairwise listing of the partitions of 2k into two parts, (s,t), with 0 < s <= t ordered by increasing values of s and where k = 1,2,... .

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A342913 Pairwise listing of the partitions of 2k into two parts, (s,t), with 0 < t <= s ordered by decreasing values of s and where k = 1,2,... .

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A102515 a(n) = floor(1 + sqrt(2n + 1)).

Views

Author

Keywords

Comments

Programs

Mathematica

Formula

Extensions

A349523 a(n) = Sum_{k=1..n} A339399(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula