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.
The first 5 terms of A083771 are the first 5 primes, so they all require a single primality test, hence the first 5 terms of this sequence are 1. For the 6th term in A083771 we must test 13, 17 until we finally find that 19 works, hence the 6th term of this sequence is 3.
k = 1; cp = 2; ct = 1; n[ct] = 2; While[ct < 64, k++; p = Prime[k]; cp1 = cp*p; If[PrimeQ[cp1 + 1], cp = cp1; ct++; n[ct] = p]]; Table[n[k], {k, 1, ct}] (Lei Zhou)
f[s_List] := Block[{p = Times @@ s, q = NextPrime@ s[[-1]]}, While[ !PrimeQ[p*q + 1], q = NextPrime@ q]; Append[s, q]]; Nest[f, {2}, 63] (* Robert G. Wilson v, Jul 20 2017 *)
2+1=3, 2*6+1=13, 2*6*8+1=97, 2*6*8*12+1=1153, etc. are primes. After 200 terms the prime is 224198929826405912196464851358435330956778558123234657623126\ 069546460095464785674042966210907411841359152393200850271694\ 899718487202330385432243578646330245831108247815285116235792\ 875886417750289946171599027675234787802312202111702704952223\ 563058999855839876391430601719636148884060097930252529666254\ 756431522481046758186320659298713737639441014068272279177710\ 551232067814381240340990584869121776471244800000000000000000\ 00000000000000000000000000000 (449 digits). - _Robert Israel_, Dec 21 2012
N := 200: # number of terms desired
P := 2:
a[1] := 2:
C := {seq(2*j, j = 2 .. 10)}:
Cmax := 20:
for n from 2 to N do
for t in C do
if isprime(t*P+1) then
a[n]:= t;
P:= t*P;
C:= C minus {t};
break;
end if;
end do;
while not assigned(a[n]) do
t0:= Cmax+2;
Cmax:= 2*Cmax;
C:= C union {seq(j, j=t0 .. Cmax, 2)};
for t from t0 to Cmax by 2 do
if isprime(t*P+1) then
a[n]:= t;
P:= t*P;
C:= C minus {t};
break;
end if
end do;
end do;
end do;
[seq(a[n],n=1..N)];
f[s_List] := Block[{k = 2, p = Times @@ s}, While[ MemberQ[s, k] || !PrimeQ[k*p + 1], k += 2]; Append[s, k]]; Nest[f, {2}, 62] (* Robert G. Wilson v, Dec 24 2012 *)
2 - 1 = 1 and 2 + 1 = 3 are both noncomposite numbers. 2*3 - 1 = 5 and 2*3 + 1 = 7 are both noncomposite numbers. 2*3*5 - 1 = 29 and 2*3*5 + 1 = 31 are both noncomposite numbers.
P:= {seq(ithprime(i),i=2..10^5)}:
R:= 2: s:= 2:
for i from 2 to 100 do
found:= false;
for p in P do
if isprime(p*s-1) and isprime(p*s+1) then R:= R,p; s:= p*s; P:= P minus {p}; found:= true; break fi;
od;
if not found then break fi
od:
R; # Robert Israel, Jan 19 2023
a[1] = 2; a[n_] := a[n] = Module[{t = Table[a[k], {k, 1, n - 1}], p = 2, r}, r = Times @@ t; While[MemberQ[t, p] || ! PrimeQ[r*p - 1] || ! PrimeQ[r*p + 1], p = NextPrime[p]]; p]; Array[a, 50]
a[1] = 3; a[n_] := a[n] = Block[{k = m = a[n - 1] - 1}, k *= 2; While[ ! PrimeQ[k + 1] || ! SquareFreeQ[k], k += m]; k + 1]; Table[(a[n + 1] - 1)/(a[n] - 1), {n, 60}] (* Stefan Steinerberger, Apr 03 2006 *)
f[s_List] := Block[{p = Times @@ s, q = 2}, While[ MemberQ[s, q] || !PrimeQ[p*q - 1], q = NextPrime@ q]; Append[s, q]]; Nest[f, {3}, 40]
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