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A249114 Take the counting numbers and continue adding 1, 2, ..., a(n) until one reaches a fourth prime.

%I A249114 #25 Oct 03 2020 09:28:21
%S A249114 7,6,19,10,12,25,11,9,40,13,15,25,11,17,67,6,15,22,15,18,43,9,12,34,
%T A249114 12,9,31,9,32,58,8,21,28,14,12,37,11,9,55,13,23,46,11,14,43,10,15,34,
%U A249114 24,26,28,9,15,37,23,18,40,6,24,61,8,18,43,22,27,37,20,9
%N A249114 Take the counting numbers and continue adding 1, 2, ..., a(n) until one reaches a fourth prime.
%C A249114 Conjecturally (Hardy & Littlewood conjecture F), a(n) exists for all n. - _Charles R Greathouse IV_, Oct 21 2014
%C A249114 It appears that the minimum value reached by a(n) is 6. This occurs for n=2, 16, 58, 136, 178, 418, 598, 808, ... . - _Michel Marcus_, Oct 26 2014
%C A249114 The conjecture in the previous line is true - if n is odd, then n+1 is even, n+3 is even, n+6 and n+10 are odd, etc., so a(n)>6. If n is even, then +1 and +3 are odd, +6, +10 are even, so the fourth prime can be first for a(n)=6. - _Jon Perry_, Oct 29 2014
%C A249114 Conjecture: a(n) is odd approximately 50% of the time. - _Jon Perry_, Oct 29 2014
%H A249114 Charles R Greathouse IV, <a href="/A249114/b249114.txt">Table of n, a(n) for n = 1..10000</a>
%F A249114 a(n) = Min_{k>0 | { n+A000217(j), j=1...k} contains four primes}. - _M. F. Hasler_, Oct 29 2014
%e A249114 a(1) = 7 because 1+1+2+3+4+5+6+7 = 29 and exactly three partial sums are prime (2,7,11).
%e A249114 a(2) = 6 because 2+1+2+3+4+5+6 = 23 and exactly three partial sums are prime (3,5,17).
%p A249114 f:= proc(n) local j,count;
%p A249114       count:= 0;
%p A249114       for j from 1 do
%p A249114         if isprime(n + j*(j+1)/2) then
%p A249114            count:= count+1;
%p A249114            if count = 4 then return j fi
%p A249114         fi
%p A249114       od
%p A249114 end proc:
%p A249114 seq(f(n),n=1..100); # _Robert Israel_, Oct 29 2014
%t A249114 a[n_] := Module[{j, cnt = 0}, For[j = 1, True, j++, If[PrimeQ[n+j(j+1)/2], cnt++; If[cnt == 4, Return[j]]]]];
%t A249114 Array[a, 100] (* _Jean-François Alcover_, Oct 03 2020, after Maple *)
%o A249114 (PARI) a(n)=my(k, s=4); while(s, if(isprime(n+=k++), s--)); k \\ _Charles R Greathouse IV_, Oct 21 2014
%Y A249114 Cf. A085415, A249112, A249113.
%K A249114 easy,nonn
%O A249114 1,1
%A A249114 _Gil Broussard_, Oct 21 2014