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

1,2

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

Cf. A096379, A138042.

Cf. A066495.

Sequence in context: A307192 A147207 A146877 * A233435 A049742 A347209

Adjacent sequences: A309717 A309718 A309719 * A309721 A309722 A309723

Keyword

nonn

Author

Philip Mizzi, Aug 14 2019

Extensions

More terms from Michel Marcus, Aug 14 2019

Status

approved