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

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

Offset: 1

Author

Gil Broussard, Oct 21 2014

Keywords

• easy
• nonn

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

Examples

			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).
		

Links

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

Crossrefs

Cf. A085415, A249112, A249113.

Programs

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] (* Jean-François Alcover, Oct 03 2020, after Maple *)

PARI

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

Formula

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