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 1-3 of 3 results.

A280252 Sum of the parts in the partitions of 2n into two squarefree parts.

Original entry on oeis.org

2, 8, 12, 24, 20, 48, 42, 80, 72, 120, 110, 168, 130, 196, 150, 256, 238, 396, 266, 440, 336, 572, 368, 624, 400, 728, 540, 728, 638, 900, 682, 960, 726, 1224, 910, 1512, 1036, 1520, 1014, 1600, 1066, 1848, 1204, 2024, 1530, 2116, 1598, 2304, 1666, 2500, 1836, 2704
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 29 2016

Keywords

Links

Crossrefs

Cf. A008683, A226237, A280226, A280250, A280251.

Programs

  • Maple
    with(numtheory): A280252:=n->2*n*add(mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280252(n), n=1..100);
  • Mathematica
    Table[2 n*Sum[MoebiusMu[i]^2 MoebiusMu[2 n - i]^2, {i, n}], {n, 80}] (* Wesley Ivan Hurt, Jan 05 2024 *)

Formula

a(n) = 2*n * A280226(n).
a(n) = A280250(n) + A280251(n).

A187619 Sum of the differences of the parts in each Goldbach partition of 2n, A187129(n) - A185297(n).

Original entry on oeis.org

0, 0, 2, 4, 2, 8, 16, 12, 20, 28, 26, 32, 24, 28, 32, 64, 60, 24, 58, 72, 86, 88, 122, 116, 78, 128, 98, 108, 144, 80, 202, 204, 60, 184, 216, 188, 226, 292, 168, 196, 316, 260, 168, 376, 236, 216, 334, 120, 304, 408, 278, 340, 472, 392, 454, 604, 452, 372, 724, 216, 330, 580, 162, 472, 542, 392, 366, 540, 470, 592, 838, 384, 390, 828
Offset: 2

Views

Author

N. J. A. Sloane, Mar 12 2011

Keywords

Links

Crossrefs

Cf. A226237 (Sum of sums), A045917.

Programs

  • Maple
    with(numtheory):
A279725:=n->2*add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (n-i), i=3..n):
seq(A279725(n), n=1..100); # Wesley Ivan Hurt, Dec 17 2016
  • Mathematica
    Table[2 Sum[(n - i) Floor[2/PrimeOmega[2 n*i - i^2]], {i, 2, n}], {n, 2, 100}] (* Wesley Ivan Hurt, Dec 20 2013 *)

Formula

a(n) = 2 * Sum_{i=2..n} (n-i) * A064911(2*n*i-i^2). - Wesley Ivan Hurt, Dec 20 2013
a(n) = 2 * Sum_{i=3..n} c(i) * c(2*n-i) * (n-i), where c = A010051. - Wesley Ivan Hurt, Dec 17 2016

Extensions

More descriptive name by Wesley Ivan Hurt, Dec 20 2013

A293909 Number of Goldbach partitions (p,q) of 2n, p <= q, such that both 2n-2 and 2n+2 have a Goldbach partition with a greater difference between its prime parts than q-p.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 3, 1, 3, 3, 2, 2, 4, 2, 3, 5, 3, 2, 5, 2, 3, 6, 2, 4, 5, 2, 4, 6, 4, 4, 6, 4, 4, 8, 4, 3, 9, 3, 4, 4, 3, 3, 8, 4, 5, 8, 5, 6, 10, 5, 5, 10, 4, 4, 8, 3, 5, 9, 5, 4, 8, 6, 7, 10, 5, 5, 11, 3, 7, 10, 5, 7, 9, 5, 5, 13, 8, 5
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 19 2017

Keywords

Examples

			a(9) = 2; Both 2(9)-2 = 16 and 2(9)+2 = 20 have two Goldbach partitions: 16 = 13+3 = 11+5 and 20 = 17+3 = 13+7. Note that 13-3 = 10 and 17-3 = 14 are the largest differences of the primes among the Goldbach partitions of 2n-2 and 2n+2. The Goldbach partitions of 2(9) = 18 are 13+5 = 11+7. Since 13-5 = 8 and 11-7 = 4 are both less than min(10,14) = 10, a(9) = 2.

Links

Crossrefs

Cf. A002375, A226237, A278700, A279103, A279315, A279481, A279727, A279728, A279729, A279792.

Extensions

More terms from Bert Dobbelaere, Sep 15 2019
Showing 1-3 of 3 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.