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A262744 Remainder when sum of first n primes is divided by n-th triangular number.

%I A262744 #22 Aug 02 2020 11:42:59
%S A262744 0,2,4,7,13,20,2,5,10,19,28,41,56,71,88,109,134,159,188,9,19,32,46,63,
%T A262744 85,108,130,153,175,198,232,267,305,342,386,429,475,524,574,627,683,
%U A262744 738,800,861,923,984,1054,1133,1213,17,46,77,106,141,178
%N A262744 Remainder when sum of first n primes is divided by n-th triangular number.
%C A262744 Sequence is interesting because a(n)-a(n-1) < 0 in certain points such as n=7 and n=20, although a(n)-a(n-1) > 0 for other points, generally.
%C A262744 Old name was: a(n) = (Sum_{k=1..n} prime(k)) mod (Sum_{k=1..n} k).
%H A262744 Alois P. Heinz, <a href="/A262744/b262744.txt">Table of n, a(n) for n = 1..10000</a>
%F A262744 a(n) = (Sum_{k=1..n} prime(k)) mod (n*(n+1)/2).
%F A262744 a(n) = A007504(n) mod A000217(n).
%e A262744 a(1) = prime(1) mod 1 = 0.
%e A262744 a(2) = (prime(1) + prime(2)) mod (1+2) = 2.
%e A262744 a(3) = (prime(1) + prime(2) + prime(3)) mod (1+2+3) = 4.
%e A262744 a(4) = (prime(1) + prime(2) + prime(3) + prime(4)) mod (1+2+3+4) = 7.
%p A262744 s:= proc(n) option remember; ithprime(n)+`if`(n>1, s(n-1), 0) end:
%p A262744 a:= n-> irem(s(n), n*(n+1)/2):
%p A262744 seq(a(n), n=1..70);  # _Alois P. Heinz_, Oct 01 2015
%t A262744 Table[Mod[Sum[Prime@ k, {k, n}], Sum[k, {k, n}]], {n, 60}] (* _Michael De Vlieger_, Sep 30 2015 *)
%t A262744 Module[{nn=60,pr,tr},pr=Accumulate[Prime[Range[nn]]];tr=Accumulate[ Range[ nn]];Mod[#[[1]],#[[2]]]&/@Thread[{pr,tr}]] (* _Harvey P. Dale_, Aug 02 2020 *)
%o A262744 (PARI) a(n) = sum(k=1, n, prime(k)) % (n*(n+1)/2);
%o A262744 vector(500, n, a(n))
%Y A262744 Cf. A007504, A000217, A090396.
%K A262744 nonn,easy,look
%O A262744 1,2
%A A262744 _Altug Alkan_, Sep 29 2015
%E A262744 New name from _Altug Alkan_, Feb 06 2017, following a suggestion from _N. J. A. Sloane_