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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 (list; graph; refs; listen; history; text; internal format)
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

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 A322049 A163260

Adjacent sequences:  A249111 A249112 A249113 * A249115 A249116 A249117

KEYWORD

easy,nonn

AUTHOR

Gil Broussard, Oct 21 2014

STATUS

approved

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Last modified March 29 05:39 EDT 2020. Contains 333105 sequences. (Running on oeis4.)