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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.

%I A025129 #37 Feb 01 2021 22:15:30
%S A025129 0,6,10,29,43,94,128,231,279,484,584,903,1051,1552,1796,2489,2823,
%T A025129 3784,4172,5515,6091,7758,8404,10575,11395,14076,15174,18339,19667,
%U A025129 23414,24906,29437,31089,36500,38614,44731,47071,54198,56914,65051,68371,77402,81052,91341
%N 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.
%C A025129 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
%H A025129 Reinhard Zumkeller, <a href="/A025129/b025129.txt">Table of n, a(n) for n = 1..10000</a>
%H A025129 Gus Wiseman, <a href="/A025129/a025129.txt">Sum of prime(i) * prime(j) for i + j = n, i != j.</a>
%F A025129 a(n) = A024697(n) for even n. - _M. F. Hasler_, Apr 06 2014
%e A025129 From _Gus Wiseman_, Dec 05 2020: (Start)
%e A025129 The sequence of sums begins (n > 1):
%e A025129     6 =  6
%e A025129    10 = 10
%e A025129    29 = 14 + 15
%e A025129    43 = 22 + 21
%e A025129    94 = 26 + 33 + 35
%e A025129   128 = 34 + 39 + 55
%e A025129   231 = 38 + 51 + 65 + 77
%e A025129   279 = 46 + 57 + 85 + 91
%e A025129 (End)
%t A025129 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 *)
%o A025129 (PARI) A025129=n->sum(k=1,n\2,prime(k)*prime(n-k+1)) \\ _M. F. Hasler_, Apr 06 2014
%o A025129 (Haskell)
%o A025129 a025129 n = a025129_list !! (n-1)
%o A025129 a025129_list= f (tail a000040_list) [head a000040_list] 1 where
%o A025129    f (p:ps) qs k = sum (take (div k 2) $ zipWith (*) qs $ reverse qs) :
%o A025129                    f ps (p : qs) (k + 1)
%o A025129 -- _Reinhard Zumkeller_, Apr 07 2014
%Y A025129 Cf. A000040, A258323.
%Y A025129 The nonsquarefree version is A024697 (shifted right).
%Y A025129 Row sums of A338905 (shifted right).
%Y A025129 A332765 is the greatest among these squarefree semiprimes.
%Y A025129 A001358 lists semiprimes.
%Y A025129 A006881 lists squarefree semiprimes.
%Y A025129 A014342 is the self-convolution of the primes.
%Y A025129 A056239 is the sum of prime indices of n.
%Y A025129 A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
%Y A025129 A339194 sums squarefree semiprimes grouped by greater prime factor.
%Y A025129 Cf. A001221, A005117, A062198, A098350, A168472, A320656, A338900, A338901, A338904, A339114, A339116.
%K A025129 nonn
%O A025129 1,2
%A A025129 _Clark Kimberling_
%E A025129 Following suggestions by _Robert Israel_ and _N. J. A. Sloane_, initial 0=a(1) added by _M. F. Hasler_, Apr 06 2014