A291564 -id:A291564 - cp's OEIS frontend

A291563 Number of partitions of 2n into two prime parts or two nonprime parts.

1, 1, 1, 2, 4, 3, 4, 6, 6, 6, 8, 9, 9, 9, 11, 9, 13, 15, 10, 14, 16, 14, 16, 19, 18, 17, 21, 18, 20, 25, 18, 24, 27, 19, 26, 28, 25, 27, 32, 26, 28, 35, 29, 29, 39, 30, 32, 38, 30, 37, 41, 35, 37, 42, 38, 41, 47, 40, 40, 54, 38, 42, 53, 39, 48, 52, 46, 46

Offset: 1

Wesley Ivan Hurt, Oct 20 2017

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Index entries for sequences related to partitions

Cf. A010051, A291564.

a:= n-> n-add(`if`(isprime(n+i) xor isprime(n-i), 1, 0), i=1..n-1):
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 05 2021

Table[Sum[KroneckerDelta[(PrimePi[k] - PrimePi[k - 1]), (PrimePi[2 n - k] - PrimePi[2 n - 1 - k])], {k, n}], {n, 80}]

a(n) = Sum_{i=1..n} [A010051(i) = A010051(2n-i)], where [] is the Iverson bracket.

a(n) = n - A291564(n).

A342155 Number of partitions of [2n] into pairs such that either their sum or their absolute difference is a prime (but not both).

Original entry on oeis.org

1, 1, 2, 3, 7, 26, 55, 282, 1520, 2685, 27005, 171474, 768123, 5936728, 43976303, 207493790, 2570789335, 21669733984, 136340261314, 1639978185920

Offset: 0

Alois P. Heinz, Mar 02 2021

			a(4) = 7:
  {{1,8}, {2,7}, {3,5}, {4,6}},
  {{1,8}, {2,7}, {3,4}, {5,6}},
  {{1,8}, {2,4}, {3,6}, {5,7}},
  {{1,8}, {2,3}, {4,6}, {5,7}},
  {{1,8}, {2,4}, {3,5}, {6,7}},
  {{1,3}, {2,4}, {5,7}, {6,8}},
  {{1,2}, {3,4}, {5,7}, {6,8}}.

Wikipedia, Partition of a set

Cf. A000040, A000341, A009692, A291564, A342136, A342139.

b:= proc(s) option remember; `if`(s={}, 1, (j-> add(`if`(i b({$1..2*n}):
seq(a(n), n=0..15);

b[s_] := b[s] = If[s == {}, 1, With[{j = Max[s]}, Sum[If[i < j && (PrimeQ[j + i] ~Xor~ PrimeQ[j - i]), b[s ~Complement~  {i, j}], 0], {i, s}]]];
a[n_] := b[Range[2n]];
a /@ Range[0, 15] (* Jean-François Alcover, Aug 25 2021, after Alois P. Heinz *)

A306878 Number of 0 < k < n such that n-k and n+k are both nonprimes.

Original entry on oeis.org

0, 0, 0, 0, 2, 1, 2, 3, 3, 3, 5, 5, 6, 6, 7, 6, 9, 10, 8, 10, 11, 10, 12, 13, 13, 13, 15, 14, 16, 18, 15, 18, 20, 16, 20, 21, 20, 21, 24, 21, 23, 26, 24, 24, 29, 25, 27, 30, 26, 30, 32, 29, 31, 33, 31, 33, 36, 33, 34, 41, 34, 36, 42, 35, 40, 42, 40, 40, 43, 42, 44, 48, 44, 44

Offset: 1

Gionata Neri, Mar 14 2019

nonn

Cf. A061357, A291564.

a(n)=my(s); for(c=1, n-1, s+=(bigomega(n-c)-1)*(bigomega(n+c)-1)!=0); s

a(n) = n - 1 - A061357(n) - A291564(n).

cp's OEIS Frontend

A291563 Number of partitions of 2n into two prime parts or two nonprime parts.

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A342155 Number of partitions of [2n] into pairs such that either their sum or their absolute difference is a prime (but not both).

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A306878 Number of 0 < k < n such that n-k and n+k are both nonprimes.

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