A309720 Numbers of the form p+q-r = q+r-s where p < q < r < s are consecutive primes.

1, 13, 37, 65, 223, 343, 637, 1087, 1273, 1423, 1445, 1483, 1603, 1687, 1867, 2077, 2135, 2375, 2605, 2683, 2705, 3029, 3523, 3545, 3913, 3997, 4123, 4633, 4783, 4927, 5435, 5735, 6079, 7205, 7295, 7331, 7547, 7589, 7811, 8159, 8227, 8701, 8827, 9085, 9335, 9457, 9461, 9923, 10057

Offset: 1

Author

Philip Mizzi, Aug 14 2019

Keywords

• nonn

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

Examples

			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.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A096379, A138042.

Cf. A066495.

Programs

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 *)

Extensions

More terms from Michel Marcus, Aug 14 2019