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A135376 a(n) is the smallest prime that does not divide n(n+1)/2.

%I A135376 #21 Dec 28 2023 12:47:55
%S A135376 2,2,5,3,2,2,3,5,2,2,5,5,2,2,7,3,2,2,3,11,2,2,5,7,2,2,5,3,2,2,3,5,2,2,
%T A135376 11,5,2,2,7,3,2,2,3,7,2,2,5,5,2,2,5,3,2,2,3,5,2,2,7,7,2,2,5,3,2,2,3,5,
%U A135376 2,2,5,5,2,2,7,3,2,2,3,7,2,2,5,11,2,2,5,3,2,2,3,5,2,2,7,5,2,2,7,3,2,2,3
%N A135376 a(n) is the smallest prime that does not divide n(n+1)/2.
%C A135376 The sums of the first 10^k terms, for k = 1, 2, ..., are 28, 354, 3596, 36026, 360402, 3604134, 36041392, 360413970, 3604140072, 36041400856, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3.604140... . - _Amiram Eldar_, Sep 10 2022
%H A135376 Harvey P. Dale, <a href="/A135376/b135376.txt">Table of n, a(n) for n = 1..1000</a>
%F A135376 a(4n+1) = a(4n+2) = 2 for all nonnegative integers n.
%F A135376 a(n) = A053670(n) for all n congruent to 0 or 3 (mod 4).
%F A135376 a(n) = A053669(A000217(n)). - _R. J. Mathar_, Dec 11 2007
%e A135376 The 11th triangular number is 66 = 2*3*11. 5 is the smallest prime that is coprime to 66, so a(11) = 5.
%p A135376 A135376 := proc(n) local T,p ; T := n*(n+1)/2 ; p := 2 ; while T mod p = 0 do p := nextprime(p) ; od: RETURN(p) ; end: seq(A135376(n),n=1..120) ; # _R. J. Mathar_, Dec 11 2007
%t A135376 a = {}; For[n = 1, n < 80, n++, j = 1; While[Mod[n*(n + 1)/2, Prime[j]] == 0, j++ ]; AppendTo[a, Prime[j]]]; a (* _Stefan Steinerberger_, Dec 10 2007 *)
%t A135376 sp[n_]:=Module[{p=2},While[Mod[n,p]==0,p=NextPrime[p]];p]; sp[#]&/@ Accumulate[ Range[110]] (* _Harvey P. Dale_, Jul 26 2018 *)
%Y A135376 Cf. A000217, A053669, A053670.
%K A135376 nonn
%O A135376 1,1
%A A135376 _Leroy Quet_, Dec 09 2007
%E A135376 More terms from _Stefan Steinerberger_ and _R. J. Mathar_, Dec 10 2007