A138042 -id:A138042 - cp's OEIS frontend

A066495 Numbers k such that f(k) = f(k-1) + f(k-2) where f denotes the prime gaps function given by f(m) = prime(m+1) - prime(m).

4, 9, 15, 21, 51, 71, 118, 184, 208, 227, 231, 238, 255, 267, 290, 317, 326, 354, 381, 392, 396, 437, 494, 499, 544, 553, 569, 627, 645, 660, 720, 756, 796, 922, 932, 937, 960, 968, 990, 1027, 1034, 1087, 1103, 1130, 1157, 1173, 1175, 1227, 1237, 1251

Offset: 1

Joseph L. Pe, Jan 03 2002

nonn

			f(9) = 6 = 4 + 2 = f(8) + f(7); so 9 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040 (function p in the definition).
Cf. A001223 (function f in the definition).
Cf. also A109226, A138042, A227419.

f[n_] := Prime[n + 1] - Prime[n]; Select[Range[3, 10^4], f[ # ] == f[ # - 1] + f[ # - 2] &]

a(n) = A138042(n) + 2 [based on the formula found from A138042]. - Antti Karttunen, Jul 13 2013

Extended by Ray Chandler, Aug 23 2005

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

Original entry on oeis.org

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

Philip Mizzi, Aug 14 2019

nonn

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

			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.

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

Cf. A096379, A138042.

Cf. A066495.

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 *)

More terms from Michel Marcus, Aug 14 2019

cp's OEIS Frontend

A066495 Numbers k such that f(k) = f(k-1) + f(k-2) where f denotes the prime gaps function given by f(m) = prime(m+1) - prime(m).

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