OFFSET
1,2
COMMENTS
The consecutive primes (p,q,r,s) satisfy 2*(r-p) = s-p. Define (p,q,r,s) = (p,p+dq,p+dr,p+ds), then 2*dr = ds. For n > 1, (r-p) == 0 (mod 6). - A.H.M. Smeets, Aug 17 2019
Correspond to where prime(i) - (prime(i+2)-prime(i+1)) values repeat. For example, 13 is obtained via both 19 - (29-23) and 17 - (23-19). - Bill McEachen, Jan 03 2021
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
EXAMPLE
Consider 4 consecutive primes (3,5,7,11), 3+5-7 = 1 = 5+7-11. 1 is a member of the sequence.
Consider 4 consecutive primes (59,61,67,71), 59+61-67 = 53 but, 61+67-71 = 57. These two sums are not equal so neither number is part of the sequence.
MATHEMATICA
upto[n_]:=Block[{p, q, r, s, t, v}, Union[ Reap[ Do[ {p, q, r, s}=t; v=p+q-r; If[ v==q+r-s <= n, Sow@ v], {t, Partition[ Prime[ Range[ 4+ PrimePi[ 2*n] ]], 4, 1]}]] [[2, 1]]]]; upto[11000] (* Giovanni Resta, Sep 06 2019 *)
#[[1]]+#[[2]]-#[[3]]&/@Select[Partition[Prime[Range[2000]], 4, 1], #[[1]]+#[[2]]- #[[3]] == #[[2]]+#[[3]]-#[[4]]&] (* Harvey P. Dale, Sep 21 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Philip Mizzi, Aug 14 2019
EXTENSIONS
More terms from Michel Marcus, Aug 14 2019
STATUS
approved