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A086447 a(n) = the least k such that prime(n+1)+prime(n+2)+...+prime(n+k) is a multiple of prime(n).

%I A086447 #21 Jul 18 2019 05:09:09
%S A086447 2,2,6,6,6,6,4,8,4,30,7,31,37,67,22,60,46,38,69,13,65,76,19,163,9,52,
%T A086447 100,84,66,136,66,119,33,79,47,76,187,214,37,96,461,111,62,189,510,37,
%U A086447 256,121,130,132,144,481,64,195,53,47,136,90,194,318,526,151,788,1542
%N A086447 a(n) = the least k such that prime(n+1)+prime(n+2)+...+prime(n+k) is a multiple of prime(n).
%C A086447 Conjecture: a(n) exists for every n.
%H A086447 Charles R Greathouse IV, <a href="/A086447/b086447.txt">Table of n, a(n) for n = 1..10000</a>
%e A086447 a(3)=6 because prime(3)=5 divides 7+11+13+17+19+23 = 90.
%t A086447 bb={}; Do[s0=Prime[n0]; s=0; Do[s+=Prime[n]; If[IntegerQ[s/s0], bb=Append[bb, n-n0]; Break[]], {n, n0+1, 8000}], {n0, 1, 100}]; bb
%t A086447 sncp[n_]:=Module[{p=Prime[n],k=n+1,t},t=Prime[k];While[!Divisible[ t, p], k++;t=t+Prime[k]];k-n]; Array[sncp,100]  (* _Harvey P. Dale_, May 21 2017 *)
%o A086447 (PARI) a(n)=my(p = prime(n), sp = nextprime(p+1), lp = sp, nb = 1); while (sp % p, lp = nextprime(lp+1); nb++; sp += lp); nb; \\ _Michel Marcus_, May 21 2017
%o A086447 (PARI) a(n, p=prime(n))=my(s, k); forprime(q=p+1, , s+=q; k++; if(s%p==0, return(k))) \\ _Charles R Greathouse IV_, May 21 2017
%Y A086447 Cf. A055233, A055514, A086448.
%K A086447 easy,nonn
%O A086447 1,1
%A A086447 _Zak Seidov_, Jul 20 2003
%E A086447 Edited by _Don Reble_, Nov 10 2005