A025129 a(n) = p(1)p(n) + p(2)p(n-1) + ... + p(k)p(n-k+1), where k = [ n/2 ], p = A000040, the primes.

0, 6, 10, 29, 43, 94, 128, 231, 279, 484, 584, 903, 1051, 1552, 1796, 2489, 2823, 3784, 4172, 5515, 6091, 7758, 8404, 10575, 11395, 14076, 15174, 18339, 19667, 23414, 24906, 29437, 31089, 36500, 38614, 44731, 47071, 54198, 56914, 65051, 68371, 77402, 81052, 91341

Offset

1,2

Comments

This is the sum of distinct squarefree semiprimes with prime indices summing to n + 1. A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798. - Gus Wiseman, Dec 05 2020

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Gus Wiseman, Sum of prime(i) * prime(j) for i + j = n, i != j.

Formula

a(n) = A024697(n) for even n. - M. F. Hasler, Apr 06 2014

Example

From Gus Wiseman, Dec 05 2020: (Start)
The sequence of sums begins (n > 1):
    6 =  6
   10 = 10
   29 = 14 + 15
   43 = 22 + 21
   94 = 26 + 33 + 35
  128 = 34 + 39 + 55
  231 = 38 + 51 + 65 + 77
  279 = 46 + 57 + 85 + 91
(End)

Mathematica

f[n_] := Block[{primeList = Prime@ Range@ n}, Total[ Take[ primeList, Floor[n/2]]*Reverse@ Take[ primeList, {Floor[(n + 3)/2], n}]]]; Array[f, 44] (* Robert G. Wilson v, Apr 07 2014 *)

Prog

(PARI) A025129=n->sum(k=1, n\2, prime(k)*prime(n-k+1)) \\ M. F. Hasler, Apr 06 2014
(Haskell)
a025129 n = a025129_list !! (n-1)
a025129_list= f (tail a000040_list) [head a000040_list] 1 where
   f (p:ps) qs k = sum (take (div k 2) $ zipWith (*) qs $ reverse qs) :
                   f ps (p : qs) (k + 1)
-- Reinhard Zumkeller, Apr 07 2014

Crossrefs

Cf. A000040, A258323.

The nonsquarefree version is A024697 (shifted right).

Row sums of A338905 (shifted right).

A332765 is the greatest among these squarefree semiprimes.

A001358 lists semiprimes.

A006881 lists squarefree semiprimes.

A014342 is the self-convolution of the primes.

A056239 is the sum of prime indices of n.

A338899/A270650/A270652 give the prime indices of squarefree semiprimes.

A339194 sums squarefree semiprimes grouped by greater prime factor.

Cf. A001221, A005117, A062198, A098350, A168472, A320656, A338900, A338901, A338904, A339114, A339116.

Sequence in context: A240972 A349846 A103767 * A093559 A269697 A271067

Adjacent sequences: A025126 A025127 A025128 * A025130 A025131 A025132

Keyword

nonn

Author

Clark Kimberling

Extensions

Following suggestions by Robert Israel and N. J. A. Sloane, initial 0=a(1) added by M. F. Hasler, Apr 06 2014

Status

approved