This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
5 belongs to the sequence because 5 = (3 + 7)/2. Likewise 53 = (47 + 59)/2. 5 belongs to the sequence because it is a term, but not first or last, of the AP of consecutive primes (3, 5, 7). 53 belongs to the sequence because it is a term, but not first or last, of the AP of consecutive primes (47, 53, 59). 257 and 263 belong to the sequence because they are terms, but not first or last, of the AP of consecutive primes (251, 257, 263, 269).
a006562 n = a006562_list !! (n-1) a006562_list = filter ((== 1) . a010051) a075540_list -- Reinhard Zumkeller, Jan 20 2012
a006562 n = a006562_list !! (n-1) a006562_list = h a000040_list where h (p:qs@(q:r:ps)) = if 2 * q == (p + r) then q : h qs else h qs -- Reinhard Zumkeller, May 09 2013
[a: n in [1..1000] | IsPrime(a) where a is NthPrime(n)-NthPrime(n+1)+NthPrime(n+2)]; // Vincenzo Librandi, Jun 23 2016
Transpose[ Select[ Partition[ Prime[ Range[1000]], 3, 1], #[[2]] ==(#[[1]] + #[[3]])/2 &]][[2]]
p=Prime[Range[1000]]; p[[Flatten[1+Position[Differences[p, 2], 0]]]]
Prime[#]&/@SequencePosition[Differences[Prime[Range[800]]],{x_,x_}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 31 2019 *)
betwixtpr(n) = { local(c1,c2,x,y); for(x=2,n, c1=c2=0; for(y=prime(x-1)+1,prime(x)-1, if(!isprime(y),c1++); ); for(y=prime(x)+1,prime(x+1)-1, if(!isprime(y),c2++); ); if(c1==c2,print1(prime(x)",")) ) } \\ Cino Hilliard, Jan 25 2005
forprime(p=1,999, p-precprime(n-1)==nextprime(p+1)-p && print1(p",")) \\ M. F. Hasler, Jun 01 2013
is(n)=n-precprime(n-1)==nextprime(n+1)-n && isprime(n) \\ Charles R Greathouse IV, Apr 07 2016
from sympy import nextprime; p, q, r = 2, 3, 5
while q < 6000:
if 2*q == p + r: print(q, end = ", ")
p, q, r = q, r, nextprime(r) # Ya-Ping Lu, Dec 23 2021
A029707 := proc(n) numtheory[pi](A001359(n)) ; end proc: seq(A029707(n),n=1..30); # R. J. Mathar, Feb 19 2017
Select[ Range@300, PrimeQ[ Prime@# + 2] &] (* Robert G. Wilson v, Mar 11 2007 *) Flatten[Position[Flatten[Differences/@Partition[Prime[Range[100]],2,1]], 2]](* Harvey P. Dale, Jun 05 2014 *)
def A029707(n) : a = [ ] for i in (1..n) : if (nth_prime(i+1)-nth_prime(i) == 2) : a.append(i) return(a) A029707(277) # Jani Melik, May 15 2014
The prime gaps split into the following runs: (1), (2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), (2), (6), (4), ...
p:= 3: t:= 1: R:= NULL: s:= 1: count:= 0: for i from 2 while count < 100 do q:= nextprime(p); g:= q-p; p:= q; if g = t then s:= s+1 else count:= count+1; R:= R, s; t:= g; s:= 1; fi od: R; # Robert Israel, Jan 06 2021
Length/@Split[Differences[Array[Prime,100]],#1==#2&]//Most
a(1) = 251 = prime(54) = A000040(54) and prime(55) - prime(54) = prime(56)-prime(55) = 6. - _Zak Seidov_, Apr 23 2011
Select[Partition[Prime[Range[9000]],4,1],Length[Union[Differences[#]]] == 1&][[All,1]] (* Harvey P. Dale, Aug 08 2017 *)
p=2;q=3;r=5;forprime(s=7,1e4, t=s-r; if(t==r-q&&t==q-p, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Feb 14 2013
n, AP, last term 1 2 2 2 2+j 3 3 3+2j 7 4 5+6j 23 5 5+6j 29 6 7+30j 157 7 7+150j 907 8 199+210j 1669 9 199+210j 1879 10 199+210j 2089 11 110437+13860j 249037 12 110437+13860j 262897 .......................... a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.
(* This program will generate the 4 to 12 terms to use a[n_] to generate term 13 or higher, it will have a prolonged run time. *) a[n_] := Module[{i, p, found, j, df, k}, i = 1; While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > ((n - 1)*df)) && (found == 0), found = 1; Do[If[! PrimeQ[p - k*df], found = 0], {k, 1, n - 1}]]; found == 0]; p]; Table[a[i], {i, 4, 12}]
From _Gus Wiseman_, May 31 2020: (Start) The first 10 strictly increasing prime gap quartets: 17 19 23 29 41 43 47 53 79 83 89 97 107 109 113 127 227 229 233 239 281 283 293 307 311 313 317 331 347 349 353 359 349 353 359 367 379 383 389 397 (End)
wpqQ[lst_]:=Module[{diffs=Differences[lst]},diffs[[1]]Harvey P. Dale, Jun 12 2012 *)
ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-xx] (* Gus Wiseman, May 31 2020 *)
The first 10 strictly decreasing prime gap quartets: 31 37 41 43 61 67 71 73 89 97 101 103 211 223 227 229 271 277 281 283 293 307 311 313 449 457 461 463 467 479 487 491 607 613 617 619 619 631 641 643 For example, the primes (211,223,227,229) have differences (12,4,2), which are strictly decreasing, so 211 is in the sequence. The second and third term of each quadruplet are consecutive terms in A051634: this is a characteristic property of this sequence. - _M. F. Hasler_, Jun 01 2020
primes:= select(isprime,[seq(i,i=3..10000,2)]): L:= primes[2..-1]-primes[1..-2]: primes[select(t -> L[t+2] < L[t+1] and L[t+1] < L[t], [$1..nops(L)-2])]; # Robert Israel, Jun 28 2018
ReplaceList[Array[Prime,100],{_,x_,y_,z_,t_,_}/;y-x>z-y>t-z:>x] (* Gus Wiseman, May 31 2020 *)
Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,1]] (* Harvey P. Dale, Jan 12 2023 *)
Select[Partition[Prime[Range[14000000]],5,1],Length[Union[ Differences[ #]]]==1&] (* Harvey P. Dale, Jun 22 2013 *)
A059044(n,p=2,c,g,P)={forprime(q=p+1,, if(p+g!=p+=g=q-p, next, q!=P+2*g, c=3, c++>4, print1(P-2*g,",");n--||break);P=q-g);P-2*g} \\ This does not impose the gap to be 30, but it happens to be the case for the first values. - M. F. Hasler, Oct 26 2018
The odd primes are: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ... with first differences: 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ... with run-lengths: 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ... with run-lengths a(n).
Length/@Split[Length /@ Split[Differences[Select[Range[3,1000],PrimeQ]]]//Most]//Most
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