cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A352749 a(n) = pi(n) * (pi(2n-1) - pi(n-1)).

Original entry on oeis.org

0, 2, 4, 4, 6, 6, 12, 8, 12, 16, 20, 20, 24, 18, 24, 30, 35, 28, 40, 32, 40, 48, 54, 54, 54, 54, 63, 63, 70, 70, 88, 77, 77, 88, 88, 99, 120, 108, 108, 120, 130, 130, 140, 126, 140, 140, 150, 135, 150, 150, 165, 180, 192, 192, 208, 208, 224, 224, 238, 221, 234, 216, 216, 234
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 01 2022

Keywords

Comments

Number of ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.
Also the number of ordered pairs of prime numbers, (p,q) that can be made with p <= q, where p and q appear as the smaller and larger parts (respectively) of the partitions of 2n into 2 parts that contain at least 1 prime.

Examples

			a(5) = 6; there are 6 ordered pairs of prime numbers, (p,q), such that p <= 5 <= q < 10: (2,5), (2,7), (3,5), (3,7), (5,5), and (5,7).
Another interpretation for a(5): the 3 partitions of 2*5 = 10 into 2 parts containing at least one prime are 2+8 = 3+7 = 5+5. There are 6 ordered pairs of primes (p,q) that can be made with p <= q, which are the same ordered pairs in the previous example.

Links

Crossrefs

Cf. A000720 (pi), A035250, A352753, A352754.

Programs

  • Mathematica
    Table[PrimePi[n] (PrimePi[2 n - 1] - PrimePi[n - 1]), {n, 100}]
  • PARI
    a(n) = primepi(n)*(primepi(2*n-1) - primepi(n-1)); \\ Michel Marcus, Apr 01 2022

Formula

a(n) = Sum_{p <= n <= q < 2n, p,q prime} 1.
a(n) = A000720(n) * A035250(n). - Bernard Schott, Apr 02 2022

A352754 a(n) = pi(n) * Sum_{n <= q < 2n, q prime} q.

Original entry on oeis.org

0, 5, 16, 24, 36, 54, 124, 96, 164, 240, 300, 360, 432, 354, 528, 714, 833, 714, 1112, 960, 1288, 1632, 1836, 2052, 2052, 2052, 2529, 2529, 2810, 3110, 4092, 3751, 3751, 4488, 4488, 5269, 6624, 6180, 6180, 7128, 7722, 8268, 8904, 8302, 9548, 9548, 10230, 9525, 10980, 10980
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 01 2022

Keywords

Comments

Sum of the primes q from the ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.

Examples

			a(5) = 36; there are 6 ordered pairs of prime numbers, (p,q), such that p <= 5 <= q < 10: (2,5), (2,7), (3,5), (3,7), (5,5), and (5,7). The sum of the corresponding prime parts q gives 5+7+5+7+5+7 = 36.

Crossrefs

Cf. A000720 (pi), A073837, A352749, A352753, A352775, A352777.

Programs

  • Mathematica
    Table[PrimePi[n] Sum[(2 n - k) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]), {k, n}], {n, 100}]

Formula

a(n) = A000720(n) * A073837(n). - Bernard Schott, Apr 02 2022
a(n) = A352775(n) - A352753(n).

A352777 a(n) = Sum_{p <= n <= q < 2n, p,q prime} (p * q).

Original entry on oeis.org

0, 10, 40, 60, 120, 180, 527, 408, 697, 1020, 1680, 2016, 2952, 2419, 3608, 4879, 6902, 5916, 10703, 9240, 12397, 15708, 20400, 22800, 22800, 22800, 28100, 28100, 36249, 40119, 59520, 54560, 54560, 65280, 65280, 76640, 108744, 101455, 101455, 117018, 141372, 151368, 178716
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 02 2022

Keywords

Comments

Total area of all unique p X q rectangles with p,q prime such that p <= n <= q < 2n.

Examples

			a(5) = 120; the 6 unique p X q rectangles, with p,q prime such that p <= 5 <= q < 10 are: 2 X 5, 2 X 7, 3 X 5, 3 X 7, 5 X 5, and 5 X 7. The total area of all rectangles is 2*5 + 2*7 + 3*5 + 3*7 + 5*5 + 5*7 = 120.

Crossrefs

Cf. A000720 (pi), A035250, A352753, A352754, A352749.

Programs

  • Mathematica
    Table[Sum[Sum[k (2 n - i) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {k, n}], {i, n}], {n, 100}]

A352775 a(n) = pi(n) * (Sum_{n <= q < 2n, q prime} q) + (pi(2n-1) - pi(n-1)) * (Sum_{p <= n, p prime} p).

Original entry on oeis.org

0, 9, 26, 34, 56, 74, 175, 130, 215, 308, 412, 472, 596, 477, 692, 919, 1123, 946, 1497, 1268, 1673, 2094, 2436, 2652, 2652, 2652, 3229, 3229, 3713, 4013, 5372, 4871, 4871, 5768, 5768, 6709, 8594, 7953, 7953, 9098, 10102, 10648, 11714, 10831, 12358, 12358, 13510
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 02 2022

Keywords

Comments

Sum of all the parts from all ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.

Examples

			a(5) = 56; there are 6 ordered pairs of prime numbers, (p,q), such that p <= 5 <= q < 10: (2,5), (2,7), (3,5), (3,7), (5,5), and (5,7). The sum of all the parts gives 2+5+2+7+3+5+3+7+5+5+5+7 = 56.

Crossrefs

Cf. A000720 (pi), A073837, A352749, A352753, A352754, A352777.

Programs

  • Mathematica
    Table[Sum[Sum[k (PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {k, n}], {i, n}] + PrimePi[n] Sum[(2 n - k) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]), {k, n}], {n, 100}]

Formula

a(n) = A352753(n) + A352754(n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.