%I #26 Jun 16 2015 13:40:21
%S 2,7,11,29,17,103,137,131,23,149,73,317,67,181,163,127,233,487,557,97,
%T 593,367,113,199,1249,2143,47,617,263,877,19,1213,349,577,383,311,643,
%U 3,1151,331,677,2521,397,1153,1381,1601,277,157,631,433,179,373,443
%N a(n) is the smallest prime different from a(k) for k < n such that sum(a(j), j=1..n) is a multiple of the n-th prime.
%C Conjecture: Every prime belongs to this sequence.
%C Among first 10^6 terms first missing primes are {5, 13, 41, 61, 71, 89, 173, 193, 227, 229, 239, 283, 313, 337, 389, 401, 439, 449, 461, 467, 479, 499, 509, 541}; only two "fixed points" are a(1) = prime(1) = 2 and a(9) = prime(9) = 23. - _Zak Seidov_, Jun 12 2015
%C Also, among first 10^6 terms there are just 16931 "early birds": terms a(k) such that a(k) < A000040(k); first values of k are: 27, 31, 38, 48, 51, 113, 99, 51, 113, 137, 191, 240, 254, 297, 308, ... - _Zak Seidov_, Jun 12 2015
%H Zak Seidov, <a href="/A073602/b073602.txt">Table of n, a(n) for n = 1..10000</a>
%e a(5) = 17 since 2 + 7 + 11 + 29 + 17 = 66 is a multiple of 11, the fifth prime. For the smaller primes 3, 5, 13 the corresponding sums 52, 54, 62 are not multiples of 11.
%t t = {2}; Do[p = Prime[n]; i = 2; While[! Divisible[Total[t] + (y = Prime[i]), p] || MemberQ[t, y], i++]; AppendTo[t, y], {n, 2, 53}]; t (* _Jayanta Basu_, Jul 02 2013 *)
%o (PARI) vsearch(n,v)=local(j,s); j=1; s=matsize(v)[2]; while(j<=s&&n!=v[j],j++); j<=s
%o {m=54; v=[]; n=1; while(n<=m,p=2; while(vsearch(p,v)||((sum(j=1,matsize(v)[2], v[j])+p)%prime(n))>0,p=nextprime(p+1)); print1(p,","); v=concat(v,p); n++)}
%K nonn
%O 1,1
%A _Amarnath Murthy_, Aug 04 2002
%E Edited and extended by _Klaus Brockhaus_, Aug 10 2002
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