A096379 -id:A096379 - cp's OEIS frontend

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

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

A191472 a(n) = 2*prime(n+2) - prime(n+1) - prime(n).

Original entry on oeis.org

5, 6, 10, 8, 10, 8, 10, 16, 10, 14, 14, 8, 10, 16, 18, 10, 14, 14, 8, 14, 14, 16, 22, 16, 8, 10, 8, 10, 32, 22, 16, 10, 22, 14, 14, 18, 14, 16, 18, 10, 22, 14, 10, 8, 26, 36, 20, 8, 10, 16, 10, 22, 22, 18, 18, 10, 14, 14, 8, 22, 38, 22, 8, 10, 32, 26, 26, 14

Offset: 1

Zak Seidov, Aug 27 2012

nonn

For n > 2, all terms >= 8 and all even integers are possible except for 12.

			a(1) = 2*prime(3) - prime(2) - prime(1) = 2*5 - 3 - 2 = 5.

Zak Seidov, Table of n, a(n) for n = 1..1000
Charles R Greathouse IV, Record values

Cf. A096379.

ps = Prime[Range[100]]; Table[2*ps[[n+2]] - ps[[n+1]] - ps[[n]], {n, Length[ps] - 2}] (* T. D. Noe, Aug 27 2012 *)
2#[[3]]-#[[2]]-#[[1]]&/@Partition[Prime[Range[70]],3,1] (* Harvey P. Dale, Aug 10 2023 *)
ListConvolve[{2, -1, -1}, Prime[Range[100]]] (* Paolo Xausa, Jul 30 2024 *)

first(n)=my(v=vector(n),p=2,q=3,k); forprime(r=5,, if(k++>n, break); v[k]=2*r-q-p; p=q; q=r); v \\ Charles R Greathouse IV, Oct 03 2017

A098865 Primes of the form prime(n) + prime(n+1) - prime(n+2).

Original entry on oeis.org

5, 7, 11, 13, 23, 37, 41, 53, 73, 97, 101, 103, 109, 127, 137, 157, 179, 191, 223, 229, 251, 263, 269, 271, 307, 311, 353, 373, 389, 409, 419, 433, 457, 479, 487, 491, 503, 541, 563, 571, 593, 641, 647, 673, 683, 691, 701, 757, 809, 821, 853, 859, 863, 877

Offset: 1

Robert G. Wilson v, Nov 04 2004

nonn

Many primes of this form arise in several different ways, e.g. 13, 37, 223, 1087, 1423, 1483, 1867, ..., . 13 = 17+19-23 = 19+23-29.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A098761, A096379.

Union[ Select[ Table[Prime[n] + Prime[n + 1] - Prime[n + 2], {n, 155}], PrimeQ[ # ] &]]
Union[Select[#[[1]]+#[[2]]-#[[3]]&/@ Partition[Prime[ Range[20000]], 3,1],PrimeQ]]  (* Harvey P. Dale, Mar 14 2011 *)

A375097 a(n) = prime(n+2) - (prime(n) + prime(n+1))/2.

Original entry on oeis.org

3, 5, 4, 5, 4, 5, 8, 5, 7, 7, 4, 5, 8, 9, 5, 7, 7, 4, 7, 7, 8, 11, 8, 4, 5, 4, 5, 16, 11, 8, 5, 11, 7, 7, 9, 7, 8, 9, 5, 11, 7, 5, 4, 13, 18, 10, 4, 5, 8, 5, 11, 11, 9, 9, 5, 7, 7, 4, 11, 19, 11, 4, 5, 16, 13, 13, 7, 5, 8, 11, 10, 9, 7, 8, 11, 8, 10, 14, 7, 11

Offset: 2

Hugo Pfoertner, Jul 30 2024

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Cf. A024675, A096379, A191472, A305748, A364411.

Array[#3 - (#1 + #2)/2 & @@ Prime[{#, # + 1, # + 2}] &, 80, 2] (* Michael De Vlieger, Jul 30 2024 *)

a(n) = prime(n+2) - (prime(n) + prime(n+1))/2

A135275 a(n) = prime(2n-1) + prime(2n) - prime(2n+1).

Original entry on oeis.org

0, 1, 7, 13, 21, 27, 37, 41, 53, 65, 69, 75, 95, 101, 95, 121, 127, 143, 153, 161, 169, 187, 185, 207, 223, 231, 235, 251, 263, 275, 269, 305, 299, 321, 343, 345, 361, 373, 385, 391, 409, 425, 433, 445, 457, 459, 479, 493, 507, 517

Offset: 1

Cino Hilliard, Dec 02 2007

Original name was: Sum and difference of staircase primes according to the rule: bottom + top - next top.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A096379, A135274.

Table[Prime[n + 1] + Prime[n] - Prime[n + 2], {n, 1, 30}][[;; ;; 2]] (* G. C. Greubel, Oct 08 2016 *)

g(n) = forstep(x=1,n,2,y=prime(x+1)+ prime(x)- prime(x+2);print1(y","))

n=1; p=2; q=3; forprime(r=5,1e3, if(n, print1(p+q-r", ")); n=!n; p=q; q=r) \\ Charles R Greathouse IV, Oct 08 2016

We list the primes in staircase fashion as in A135274. The right diagonal, RD(n), is the set of top primes and the left diagonal, LD(x), is the set of bottom primes. Then a(n) = LD(n+1) + RD(n) - RD(n+2).

a(n) = A096379(2*n-1). - R. J. Mathar, Sep 10 2016

New name from Charles R Greathouse IV, Oct 08 2016

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

A138197 Array read by antidiagonals: T(n,k) = T(n-1,k) + T(n,k+1) - T(n,k+2).

Original entry on oeis.org

2, 3, 0, 5, 1, 0, 7, 1, -3, -2, 11, 5, -1, -5, -2, 13, 7, 1, -5, -5, 14, 17, 11, 5, -5, -21, -27, 24, 19, 13, 11, 11, 1, -37, -91, -54, 23, 13, 5, 5, 17, 27, -13, -153, -294, 29, 21, 11, -1, -9, 3, 49, 87, -57, -588, 31, 23, 17, 13, 5, -19, -51, -9, 237, 527, -156, 37, 27, 15, 7, 15, 35, 7, -159, -347, 95, 1871, 3212, 41, 35, 25, 5, -15

Offset: 1

Zak Seidov, Mar 05 2008

Rule A096379 applied recursively.

			Top left corner of array:
     2,    3,     5,     7,   11,   13,    17,    19,   23,  29, ... = A000040
     0,    1,     1,     5,    7,   11,    13,    13,   21,  23, ... = A096379
     0,   -3,    -1,     1,    5,   11,     5,    11,   17,  15, ...
    -2,   -5,    -5,    -5,   11,    5,    -1,    13,    7,   5, ...
    -2,   -5,   -21,     1,   17,   -9,     5,    15,  -15,  -9, ...
    14,  -27,   -37,    27,    3,  -19,    35,     9,  -71,  -5, ...
    24,  -91,   -13,    49,  -51,    7,   115,   -57, -167,  69, ...
   -54, -153,    87,    -9, -159,  179,   225,  -293, -217, 223, ...
  -294,  -57,   237,  -347, -205,  697,   149,  -733,  -59, 341, ...
  -588,  527,    95, -1249,  343, 1579,  -525, -1133,  335, 271, ...
  -156, 1871, -1497, -2485, 2447, 2187, -1993, -1069,  643, 261, ...
  ...

Cf. A000040, A096379.

T(0,k) = A000040(k), and T(n,k) = T(n-1,k) + T(n,k+1) - T(n,k+2).

Typo corrected by Travis Hoppe, Apr 24 2008

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

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

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A191472 a(n) = 2*prime(n+2) - prime(n+1) - prime(n).

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A098865 Primes of the form prime(n) + prime(n+1) - prime(n+2).

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A375097 a(n) = prime(n+2) - (prime(n) + prime(n+1))/2.

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A135275 a(n) = prime(2n-1) + prime(2n) - prime(2n+1).

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

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A138197 Array read by antidiagonals: T(n,k) = T(n-1,k) + T(n,k+1) - T(n,k+2).

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

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