A116994 - OEIS

%I #12 Feb 02 2019 20:28:29

%S 3,1759,3323,469303,605113,641969,1110587,1426669,11148289,18352349,

%T 20473721,21820391,24710753,30048589,36690923,40785301,97060681,

%U 155135369,160593239,168132247,361391623,377965069,416572171,645803201

%N Prime partial sums of triangular numbers with prime indices.

%H Harvey P. Dale, <a href="/A116994/b116994.txt">Table of n, a(n) for n = 1..2000</a>

%F A000040 INTERSECTION {A085739 Partial sums of A034953(n)}. Primes in A085739. (Sum_{i=1..k} A000217(A000040(i))) iff in A000040. (Sum_{i=1..k} (A000040(i)*(A000040(i)+1)/2) iff in A000040.

%e a(1) = Sum_{i=1..1} prime(i)*(prime(i)+1)/2 = T(2) = 3.

%e a(2) = Sum_{i=1..11} prime(i)*(prime(i)+1)/2 = T(2)+T(3)+T(5)+T(7)+T(11)+T(13)+T(17)+T(19)+T(23)+T(29)+T(31) = 1759.

%e a(3) = Sum_{i=1..13} prime(i)*(prime(i)+1)/2 = 3323.

%e a(4) = Sum_{i=1..53} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(241) = 469303.

%e a(5) = Sum_{i=1..57} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(269) = 605113.

%e a(6) = Sum_{i=1..58} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(271) = 641969.

%e a(7) = Sum_{i=1..68} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(337) = 1110587.

%p T:=n->n*(n+1)/2: a:=proc(n): if isprime(sum(T(ithprime(j)),j=1..n))=true then sum(T(ithprime(j)),j=1..n) else fi end: seq(a(n),n=1..500); # _Emeric Deutsch_, Apr 06 2006

%t Select[Accumulate[Table[(n(n+1))/2,{n,Prime[Range[500]]}]],PrimeQ] (* _Harvey P. Dale_, Jan 25 2015 *)

%Y Cf. A000040, A000217, A034953, A085739.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Apr 02 2006

%E More terms from _Emeric Deutsch_, Apr 06 2006