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A309720 Numbers of the form p+q-r = q+r-s where p < q < r < s are consecutive primes. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 3 13:56 EST 2022. Contains 358534 sequences. (Running on oeis4.)