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A014148 a(n) = Sum_{m=1..n} Sum_{k=1..m} prime(k).

%I A014148 #34 Jul 08 2025 02:53:05
%S A014148 2,7,17,34,62,103,161,238,338,467,627,824,1062,1343,1671,2052,2492,
%T A014148 2993,3561,4200,4912,5703,6577,7540,8600,9761,11025,12396,13876,15469,
%U A014148 17189,19040,21028,23155,25431,27858,30442,33189,36103,39190,42456,45903
%N A014148 a(n) = Sum_{m=1..n} Sum_{k=1..m} prime(k).
%C A014148 Previous name was: Apply partial sum operator twice to sequence of primes.
%C A014148 Numbers n such that a(n) is prime are listed in A122381(n) = {1, 2, 3, 6, 10, 23, 31, 46, 55, 58, 66, 70, 82, 91, 118, 131, 151, 163, 182, 187, 198, 199, ...}. Corresponding primes a(n) = a( A122381(n) ) = A122382(n) = {2, 7, 17, 103, 467, 6577, 17189, 61627, 109919, 130531, 198109, 239579, 399557, 559313, ...}. - _Alexander Adamchuk_, Aug 30 2006
%C A014148 Row 2 in A254858. - _Reinhard Zumkeller_, Feb 08 2015
%C A014148 Partial sums of A007504, n>=1. - _Omar E. Pol_, Nov 23 2016
%H A014148 Harvey P. Dale, <a href="/A014148/b014148.txt">Table of n, a(n) for n = 1..10000</a> [extending prior b-File from Alexander Adamchuk]
%F A014148 Convolution of the primes with the positive integers: Sum_{k=1..n} (n-k+1)*prime(k). - _David Scambler_, Oct 08 2006
%p A014148 b:= proc(n) option remember; `if`(n<1, [0$2],
%p A014148       (p-> p+[ithprime(n), p[1]])(b(n-1)))
%p A014148     end:
%p A014148 a:= n-> b(n+1)[2]:
%p A014148 seq(a(n), n=1..42);  # _Alois P. Heinz_, Oct 07 2021
%t A014148 Table[Sum[Sum[Prime[k],{k,1,m}],{m,1,n}],{n,1,100}] (* _Alexander Adamchuk_, Aug 30 2006 *)
%t A014148 Accumulate[Accumulate[Prime[Range[50]]]] (* _Harvey P. Dale_, Dec 29 2011 *)
%o A014148 (Haskell)
%o A014148 a014148 n = a014148_list !! (n-1)
%o A014148 a014148_list = (iterate (scanl1 (+)) a000040_list) !! 2
%o A014148 -- _Reinhard Zumkeller_, Feb 08 2015
%Y A014148 Cf. A000040, A007504, A014150, A122381, A122382, A178138, A254784, A254858.
%K A014148 nonn
%O A014148 1,1
%A A014148 _N. J. A. Sloane_
%E A014148 More terms from _Alexander Adamchuk_, Aug 30 2006
%E A014148 Name changed by _Wesley Ivan Hurt_, Oct 04 2021