A294111 -id:A294111 - cp's OEIS frontend

A368058 Sum of the smaller parts of the partitions of n into two distinct parts with larger part prime.

0, 0, 1, 1, 2, 1, 2, 4, 6, 3, 4, 6, 8, 4, 6, 8, 10, 13, 16, 20, 24, 17, 20, 24, 28, 19, 22, 25, 28, 32, 36, 41, 46, 34, 38, 42, 46, 32, 36, 40, 44, 49, 54, 60, 66, 49, 54, 60, 66, 72, 78, 84, 90, 97, 104, 111, 118, 96, 102, 109, 116, 93, 100, 107, 114, 121, 128, 136, 144, 152, 160

Offset: 1

Wesley Ivan Hurt, Dec 09 2023

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A010051, A294111, A294487, A294602.

N:= 100: # for a(1) .. a(N)
V:= Vector(N):
for i from 1 do
  p:= ithprime(i);
  if p >= N then break fi;
  m:= min(2*p-1,N);
  V[p+1..m]:= V[p+1..m] + <$1..m-p>
od:
convert(V,list); # Robert Israel, Jan 26 2024

Table[Sum[i (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}],{n, 100}]

a(n) = Sum_{i=1..floor((n-1)/2)} i * c(n - i), where c is the prime characteristic (A010051).

a(n) = n*A294602(n) - A294487(n). - Wesley Ivan Hurt, Dec 09 2023

A294109 Sum of the larger parts of the partitions of n into two parts with smaller part prime.

Original entry on oeis.org

0, 0, 0, 2, 3, 7, 9, 11, 13, 20, 23, 26, 29, 39, 43, 47, 51, 55, 59, 63, 67, 82, 87, 92, 97, 115, 121, 127, 133, 139, 145, 151, 157, 180, 187, 194, 201, 227, 235, 243, 251, 259, 267, 275, 283, 314, 323, 332, 341, 350, 359, 368, 377, 386, 395, 404, 413, 451

Offset: 1

Wesley Ivan Hurt, Oct 22 2017

Index entries for sequences related to partitions

Cf. A010051, A294111.

Table[Sum[(n - i) (PrimePi[i] - PrimePi[i - 1]), {i, Floor[n/2]}], {n, 80}]
Table[Total[Select[IntegerPartitions[n,{2}],PrimeQ[#[[2]]]&][[All,1]]],{n,80}] (* Harvey P. Dale, Jul 08 2019 *)

a(n) = Sum_{i=1..floor(n/2)} (n - i) * c(i), where c is the prime characteristic (A010051).

cp's OEIS Frontend

A368058 Sum of the smaller parts of the partitions of n into two distinct parts with larger part prime.

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