A066495 -id:A066495 - cp's OEIS frontend

A117876 Primes p=prime(k) of level (1,2), i.e., such that A118534(k) = prime(k-2).

23, 47, 73, 233, 353, 647, 1097, 1283, 1433, 1453, 1493, 1613, 1709, 1889, 2099, 2161, 2383, 2621, 2693, 2713, 3049, 3533, 3559, 3923, 4007, 4133, 4643, 4793, 4937, 5443, 5743, 6101, 7213, 7309, 7351, 7561, 7621, 7829, 8179, 8237, 8719, 8849, 9109, 9343, 9467

Offset: 1

Rémi Eismann, May 02 2006

nonn

If prime(k) has level 1 in A117563, and if 2*prime(k) - prime(k+1) = prime(k-i), then we say that prime(k) has level (1,i). Sequence gives primes of level (1,2).

The prime p(4)=7 cannot be decomposed into weight*level+gap (<=> A117563(4)=0 <=> A118534(4)=0 <=> A117078(4)=0). For all other primes, an equivalent definition would be: Primes p(k) such that 2*p(k) - p(k+1) = p(k-2). - Rémi Eismann and M. F. Hasler, Nov 08 2009

			29 = 2*23 - 17, 2179 = 2*2161 - 2143, 5749 = 2*5743 - 5737.

Remi Eismann, Table of n, a(n) for n = 1..10000

Cf. A000040, A066495, A117078, A117563, A118534.

With[{m = 2}, Prime@ Select[Range[m + 1, 1200], If[MemberQ[{1, 2, 4}, #], 0, 2 Prime[#] - Prime[# + 1]] == Prime[# - m] &]] (* Michael De Vlieger, Jul 16 2017 *)

for(n=5,9999, 2*prime(n)-prime(n+1) == prime(n-2) & print1(prime(n),",")) \\ M. F. Hasler, Nov 08 2009

is_A117876(p)={ isprime(p) & isprime(d=2*p-nextprime(p+2)) & d == precprime(precprime(p-2)-2) & p>7 } \\ M. F. Hasler, Nov 08 2009

(define (A117876 n) (A000040 (A066495 (+ 1 n)))) ;; Antti Karttunen, Nov 30 2013

a(n) = A000040(A066495(n+1)). - Antti Karttunen, Nov 30 2013

Edited by N. J. A. Sloane, May 14 2006
More terms from Rémi Eismann, May 25 2006
Definition corrected and terms double-checked by M. F. Hasler, Nov 08 2009

A138042 Numbers k such that A096379(k)=A096379(k+1).

Original entry on oeis.org

2, 7, 13, 19, 49, 69, 116, 182, 206, 225, 229, 236, 253, 265, 288, 315, 324, 352, 379, 390, 394, 435, 492, 497, 542, 551, 567, 625, 643, 658, 718, 754, 794, 920, 930, 935, 958, 966, 988, 1025, 1032, 1085, 1101, 1128, 1155, 1171, 1173, 1225, 1235, 1249

Offset: 1

Zak Seidov, Mar 02 2008

nonn

Numbers k such that prime(k)=2*prime(k+2)-prime(k+3).

			n=2: {prime(n), 2*prime(n+2)-prime(n+3)}={3,2*7-11},
n=7: {prime(7), 2*prime(9)-prime(10)}={17,2*23-29},
n=13: {prime(13), 2*prime(15)-prime(16)}={41,2*47-53},
n=19: {prime(19), 2*prime(21)-prime(22)}={67,2*73-79}.

Zak Seidov, Table of n, a(n) for n = 1..1192, a(n)<50000.

Cf. A066495, A096379.

Do[If[Prime[n]==2Prime[n+2]-Prime[n+3],Print[n]],{n,1,50000}]

;; With Antti Karttunen's IntSeq-library.
(define A138042 (MATCHING-POS 1 1 (lambda (n) (= 2 (/ (+ (A000040 n) (A000040 (+ n 3))) (A000040 (+ n 2)))))))

a(n) = A066495(n) - 2.

Formula corrected by Antti Karttunen, Jul 13 2013

A109226 If g(x) is the x-th prime gap, then g(a(n)) are prime gaps which are greater than the sum of the preceding two prime gaps.

Original entry on oeis.org

30, 34, 42, 46, 53, 61, 62, 66, 91, 97, 99, 106, 114, 121, 137, 145, 146, 150, 154, 162, 172, 175, 180, 189, 203, 214, 217, 221, 232, 239, 250, 258, 259, 263, 266, 274, 278, 289, 293, 297, 304, 309, 316, 319, 324, 331, 334, 335, 338, 342, 344, 350, 357, 360

Offset: 1

Ray G. Opao, Aug 19 2005

nonn

			34 is in the sequence because if g(34) = 35th_prime - 34th_prime = 149 - 139 = 10 and g(33) = 34th_prime - 33rd_prime = 139 - 137 = 2 and g(32) = 33rd_prime - 32nd_prime = 137 - 131 = 6, then g(34) > g(33) + g(32) or 10 > 2 + 6

Cf. A001223, A066495.

g[n_] := Prime[n + 1] - Prime[n]; Select[Range[3, 360], g[ # ] > g[ # - 1] + g[ # - 2] &] (* Ray Chandler, Aug 23 2005 *)

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

A375087 Numbers added to cumulative correction term in order for prime numbers to resemble a recursive sequence.

Original entry on oeis.org

0, 1, 0, 4, 2, 4, 2, 0, 8, 2, 4, 8, 2, 0, 4, 10, 2, 4, 8, 0, 4, 4, 2, 10, 10, 2, 4, 2, -8, 14, 12, 8, -2, 10, 6, 2, 8, 4, 4, 10, -2, 10, 8, 4, -6, 2, 20, 14, 2, 0, 8, -2, 6, 10, 6, 10, 2, 4, 8, -4, -2, 20, 16, 2, -8, 12, 10, 14, 8, 0, 2, 8, 8, 8, 4, 2, 10, 4, 2, 16, 2, 10

Offset: 1

Kaleb Williams, Jul 29 2024

At n=1, prime(n+2) = prime(n+1) + prime(n) but thereafter such a form must be reduced by a "correction" amount prime(n+2) = prime(n+1) + prime(n) - A096379(n), and the present sequence is how that correction changes.

			For n = 1: a(1) = p_2 + p_1 - p_3 - (Sum_{i <= 0} a(i)) = p_2 + p_1 - p_3 ==> a(1) = 3 + 2 - 5 = 0 ==> a(1) = 0.
For n = 2: a(2) = p_3 + p_2 - p_4 - (Sum_{i <= 1} a(i)) = p_3 + p_2 - p_4 - a(1) ==> a(2) = 5 + 3 - 7 - 0 = 1 ==> a(2) = 1.
For n = 3: a(3) = p_4 + p_3 - p_5 - (Sum_{i <= 2} a(i)) = p_4 + p_3 - p_5 - (a(1) + a(2)) ==> a(3) = 7 + 5 - 11 - (0 + 1) = 0 ==> a(3) = 0.

Cf. A096379 (partial sums), A066495 (indices of 0's).

lista(nn) = my(va = vector(nn)); for (n=1, nn, va[n] = prime(n+1) + prime(n) - prime(n+2) - sum(i=1, n-1, va[i]);); va; \\ Michel Marcus, Jul 30 2024

a(n) = 2*prime(n+1) - prime(n+2) - prime(n-1), for n>=2.
a(n) = A096379(n) - A096379(n-1), for n>=2.
prime(n+2) = prime(n+1) + prime(n) - Sum_{i=1..n} a(i)
a(n) = prime(n+1) + prime(n) - prime(n+2) - Sum_{i=0..n-1} a(i).

cp's OEIS Frontend

A117876 Primes p=prime(k) of level (1,2), i.e., such that A118534(k) = prime(k-2).

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A138042 Numbers k such that A096379(k)=A096379(k+1).

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A109226 If g(x) is the x-th prime gap, then g(a(n)) are prime gaps which are greater than the sum of the preceding two prime gaps.

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Mathematica

Extensions

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

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Mathematica

Extensions

A375087 Numbers added to cumulative correction term in order for prime numbers to resemble a recursive sequence.

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