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A143121 Triangle read by rows, T(n,k) = Sum_{j=k..n} prime(j), 1 <= k <= n.

%I A143121 #29 Dec 16 2022 09:00:10
%S A143121 2,5,3,10,8,5,17,15,12,7,28,26,23,18,11,41,39,36,31,24,13,58,56,53,48,
%T A143121 41,30,17,77,75,72,67,60,49,36,19,100,98,95,90,83,72,59,42,23,129,127,
%U A143121 124,119,112,101,88,71,52,29,160,158,155,150,143,132,119,102,83,60,31
%N A143121 Triangle read by rows, T(n,k) = Sum_{j=k..n} prime(j), 1 <= k <= n.
%C A143121 Left border = A007504, sum of first n primes: (2, 5, 10, 27, 28, 41, ...).
%C A143121 Right border = primes = A000040.
%C A143121 Row sums = A014285: (2, 8, 23, 51, 106, 184, ...).
%H A143121 G. C. Greubel, <a href="/A143121/b143121.txt">Rows n=1..100 of triangle, flattened</a>
%F A143121 T(n,k) = Sum_{j=k..n} prime(j), 1 <= k <= n, primes = A000040.
%F A143121 Equals A000012 * (A000040 * 0^(n-k)) * A000012.
%e A143121 First few rows of the triangle are:
%e A143121    2;
%e A143121    5,  3;
%e A143121   10,  8,  5;
%e A143121   17, 15, 12,  7;
%e A143121   28, 26, 23, 18, 11;
%e A143121   41, 39, 36, 31, 24, 13;
%e A143121   58, 56, 53, 48, 41, 30, 17;
%e A143121   ...
%e A143121 T(5,3) = 23 = prime(3) + prime(4) + prime(5) = (5 + 7 + 11).
%p A143121 a:=proc(n,k) add(ithprime(j),j=k..n) end: seq(seq(a(n,k),k=1..n),n=1..11); # _Muniru A Asiru_, Oct 15 2018
%t A143121 a[n_, k_] := a[n, k] = Plus@@Prime[Range[n - k + 1, n]]; Column[Table[a[n, k], {n, 15}, {k, n, 1, -1}], Center] (* _Alonso del Arte_, Jul 25 2011 *)
%t A143121 Table[Sum[Prime[j], {j, k, n}], {n, 1, 15}, {k, 1, n}]//Flatten (* _G. C. Greubel_, Oct 15 2018 *)
%o A143121 (PARI) a(n,k)=my(s);forprime(p=prime(k),prime(n),s+=p);s \\ _Charles R Greathouse IV_, Jul 25 2011
%o A143121 (Magma) [[(&+[NthPrime(j): j in [k..n]]): k in [1..n]]: n in [1..15]]; // _G. C. Greubel_, Oct 15 2018
%Y A143121 Cf. A000040, A007504, A014285, A000012.
%Y A143121 Cf. A194939 (rows reversed).
%K A143121 nonn,easy,tabl
%O A143121 1,1
%A A143121 _Gary W. Adamson_ & _Roger L. Bagula_, Jul 26 2008
%E A143121 Corrected by Hanke Bremer, Nov 28 2008