A263171 Smallest prime starting a sequence of 4 consecutive odd primes such that the center of the symmetrical gaps is 2n.

7, 5, 251, 353, 137, 2393, 109, 1931, 1753, 883, 3733, 7351, 12007, 2969, 8887, 27697, 1321, 22811, 38377, 62987, 183823, 15679, 124001, 180563, 45887, 48677, 100847, 178693, 152993, 557087, 34057, 367949, 294551, 134507, 173357, 1802407, 531359, 1134311, 933067

Offset: 1

Author

Michel Lagneau, Oct 11 2015

Keywords

• nonn

Comments

The sequence is generalizable with primes starting a sequence of 2k consecutive odd primes.

Conjecture: a(n) exists for all n>0.

Examples

			a(2)=5 because the 4 consecutive primes 5, 7, 11, 13 have gaps 2, 4, 2, which is symmetric about its center 4 = 2*2.
		

Crossrefs

Cf. A001223, A055380, A055381, A055382, A175309.

Programs

Maple

with(numtheory):nn:=500000:l:=2:T:=array(1..2*l-1)):
for n from 1 to 35 do:ii:=0:
  for k from 1 to nn while(ii=0) do:
      lst:={}:lst1:={}:
       for m from 1 to 2*l do:
        lst:=lst union {ithprime(k+m-1)}
       od:
         for p from 1 to 2*l do:
          lst1:=lst1 union {lst[p]+lst[2*l-p+1]}
         od:
            n0:=nops(lst1):
            if n0=1
            then
           for a from 1 to 2*l-1 do:
           T[a]:=lst[a+1]-lst[a]:
           od:
           if T[2]=2*n then ii:=1:printf(`%d, `,lst[1]):
           else fi :fi:
           od :
          od:

PARI

a(n) = {pa = 3; pb = 5; pc = 7; forprime(p=8, , if (((pc-pb) == 2*n) && ((pb-pa) == (p-pc)), return(pa)); pa = pb; pb = pc; pc = p;);} \\ Michel Marcus, Oct 16 2015