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A064009 a(n) = Sum_{k=1..n} d(k)*prime(k), where d(k) = A001223.

%I A064009 #38 Jun 25 2022 01:05:41
%S A064009 2,8,18,46,68,120,154,230,368,426,612,760,842,1014,1296,1614,1732,
%T A064009 2098,2366,2508,2946,3262,3760,4472,4860,5062,5474,5688,6124,7706,
%U A064009 8214,9000,9274,10664,10962,11868,12810,13462,14464,15502,15860,17670,18052
%N A064009 a(n) = Sum_{k=1..n} d(k)*prime(k), where d(k) = A001223.
%H A064009 Harry J. Smith, <a href="/A064009/b064009.txt">Table of n, a(n) for n=1..1000</a>
%e A064009 a(4) = 2*(3-2) + 3*(5-3) + 5*(7-5) + 7*(11-7) = 46.
%t A064009 Accumulate@ Array[# (NextPrime@ # - #) &@ Prime@ # &, {43}] (* _Michael De Vlieger_, May 05 2016 *)
%o A064009 (PARI) d(n) = prime(n+1)-prime(n); j=[]; for(n=1,150,j=concat(j,sum(k=1,n, prime(k)*d(k)))); j
%o A064009 (PARI) d(n)= { prime(n + 1) - prime(n) }
%o A064009 { a=0; for (n=1, 1000, write("b064009.txt", n, " ", a+=prime(n)*d(n)) ) } \\ _Harry J. Smith_, Sep 05 2009
%o A064009 (Sage) p=primes_first_n(44); d=differences(p); v=[]; a=0
%o A064009 for k in range(len(d)):
%o A064009    a+=p[k]*d[k]; v.append(a)
%o A064009 print(v)  # _Giuseppe Coppoletta_, May 05 2016
%Y A064009 Cf. A001223.
%Y A064009 Partial sums of A291463.
%K A064009 nonn
%O A064009 1,1
%A A064009 _Jason Earls_, Sep 06 2001