A294113 Sum of the smaller parts of the partitions of 2n into two parts with larger part prime.

0, 3, 4, 4, 8, 6, 11, 8, 13, 20, 28, 24, 32, 25, 32, 41, 51, 42, 51, 40, 49, 60, 72, 60, 72, 84, 97, 111, 125, 109, 124, 107, 121, 136, 152, 169, 188, 169, 187, 206, 226, 204, 224, 199, 218, 238, 258, 229, 248, 268, 289, 312, 336, 306, 331, 357, 384, 412

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..n} i * A010051(2n-i).

a(n) = 2*n*(A000720(2*n)-A000720(n-1)) - A034387(2*n) + A034387(n-1) for n >= 2. - Robert Israel, Mar 13 2018

Example

For n=7, 2n = 14 can be partitioned into two parts with the larger part prime as 13 + 1, 11 + 3, and 7 + 7. So a(7) = 1 + 3 + 7 = 11. - Michael B. Porter, Mar 14 2018

Maple

N:= 1000: # to get a(1)..a(n)
P:= select(isprime, [2, seq(i, i=3..2*N, 2)]):
S:= ListTools:-PartialSums(P):
f:= proc(n) local k1, k2;
     k1:= numtheory:-pi(2*n);
     k2:= numtheory:-pi(n-1);
     2*n*(k1-k2) - S[k1] + S[k2]
end proc:
f(1):= 0:
seq(f(n), n=1..N); # Robert Israel, Mar 13 2018

Mathematica

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

Prog

(PARI) a(n) = sum(k=1, n, k*isprime(2*n-k)); \\ Michel Marcus, Oct 24 2017
(PARI) a(n) = my(res = 0); forprime(p = n, 2*n, res+=(2*n - p)); res \\ David A. Corneth, Oct 24 2017

Crossrefs

Cf. A000720, A010051, A034387, A294114.

Sequence in context: A021033 A269714 A146944 * A339827 A240875 A127735

Adjacent sequences: A294110 A294111 A294112 * A294114 A294115 A294116

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 22 2017

Status

approved