

A249114


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


1



7, 6, 19, 10, 12, 25, 11, 9, 40, 13, 15, 25, 11, 17, 67, 6, 15, 22, 15, 18, 43, 9, 12, 34, 12, 9, 31, 9, 32, 58, 8, 21, 28, 14, 12, 37, 11, 9, 55, 13, 23, 46, 11, 14, 43, 10, 15, 34, 24, 26, 28, 9, 15, 37, 23, 18, 40, 6, 24, 61, 8, 18, 43, 22, 27, 37, 20, 9
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OFFSET

1,1


COMMENTS

Conjecturally (Hardy & Littlewood conjecture F), a(n) exists for all n.  Charles R Greathouse IV, Oct 21 2014
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
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
Conjecture: a(n) is odd approximately 50% of the time.  Jon Perry, Oct 29 2014


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


FORMULA

a(n) = Min_{k>0  { n+A000217(j), j=1...k} contains four primes}.  M. F. Hasler, Oct 29 2014


EXAMPLE

a(1) = 7 because 1+1+2+3+4+5+6+7 = 29 and exactly three partial sums are prime (2,7,11).
a(2) = 6 because 2+1+2+3+4+5+6 = 23 and exactly three partial sums are prime (3,5,17).


MAPLE

f:= proc(n) local j, count;
count:= 0;
for j from 1 do
if isprime(n + j*(j+1)/2) then
count:= count+1;
if count = 4 then return j fi
fi
od
end proc:
seq(f(n), n=1..100); # Robert Israel, Oct 29 2014


MATHEMATICA

a[n_] := Module[{j, cnt = 0}, For[j = 1, True, j++, If[PrimeQ[n+j(j+1)/2], cnt++; If[cnt == 4, Return[j]]]]];
Array[a, 100] (* JeanFrançois Alcover, Oct 03 2020, after Maple *)


PROG

(PARI) a(n)=my(k, s=4); while(s, if(isprime(n+=k++), s)); k \\ Charles R Greathouse IV, Oct 21 2014


CROSSREFS

Cf. A085415, A249112, A249113.
Sequence in context: A215334 A298377 A299244 * A275372 A338137 A341289
Adjacent sequences: A249111 A249112 A249113 * A249115 A249116 A249117


KEYWORD

easy,nonn


AUTHOR

Gil Broussard, Oct 21 2014


STATUS

approved



