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.
p:= 2: q:= 3: r:= 5: for i from 2 to 200 do t:= q - (p+r)/2; A[i]:= piecewise(t<0,1,t=0,2,3); p:= q; q:= r; r:= nextprime(r); od: seq(A[i],i=2..200); # Robert Israel, Mar 25 2018
A108415[n_]:=2+Sign[Prime[n]-1/2(Prime[n-1]+Prime[n+1])]
my(q=3, r=2, s=0); forprime(p=5,default(primelimit),(s+=sign(r+0-2*(r=q)+q=p))||print1(r, ", "))
Select[Partition[Prime[Range[750]],3,1],Round[HarmonicMean[{#[[1]],#[[3]]}]]==#[[2]]&][[;;,2]] (* Harvey P. Dale, Dec 29 2024 *)
a349793(limit) = {my(p1=2,p2=3); forprime(p3=5, limit, my(hm=round((2*p1*p3)/(p1+p3))); if(p2==hm, print1(p2,", ")); p1=p2;p2=p3)};
a349793(5500)
According to the definition in A348168, prime numbers are divided into sublists, L_1, L_2, L_3,..., in which L_n = [p(n,1), p(n,2), ..., p(n,m(n))], where p(n,k) is the k-th prime and m(n) the number of primes in the n-th sublist L_n. Thus, a(n) = p(n,1). The first sublist L_1 = [2]. If p(n,1) <= (prevprime(p(n,1)) + nextprime(p(n,1)))/2, then L_n has only 1 prime, p(n,1). Otherwise, m(n) is the largest integer such that g(n,1) >= g(n,i), where g(n,i) = p(n,i+1) - p(n,i) and 2 <= i <= m(n). The first 32 primes, for example, are divided into 16 prime sublists: [2], [3], [5], [7], [11,13], [17,19], [23], [29,31], [37,41,43,47], [53], [59,61], [67,71,73], [79,83], [89], [97,101,103,107,109,113], [127,131]. The leading primes in these sublists are: 2, 3, 5, 7, 11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127. Therefore, a(1) = 2, a(2) = 3, ..., and a(16) = 127.
from sympy import nextprime; R = [2]; L = [2]
for n in range(2, 57):
p0 = L[-1]; p1 = nextprime(p0); M = [p1]; g0 = p1-p0; p = nextprime(p1); g1 = p-p1
while g1 < g0 and p-p1 <= g1: M.append(p); p1 = p; p = nextprime(p)
L = M; R.append(L[0])
print(*R, sep =', ')
wpqQ[listn_]:=Module[{d=Differences[listn]},d[[1]]Harvey P. Dale, Apr 11 2011 *)
Transpose[Select[Partition[Prime[Range[350]],4,1],Min[ Differences[ #,2]]> 1&]][[4]] (* Harvey P. Dale, May 24 2015 *)
lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];If[p1^2<(p0^2+p2^2)/2,AppendTo[lst,p1]],{n,5!}];lst
Select[Partition[Prime[Range[100]],3,1],#[[2]]^2<(#[[1]]^2+#[[3]]^2)/2&][[All,2]] (* Harvey P. Dale, May 01 2021 *)
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