A068192 Let a(1)=2, f(n) = 4*a(1)*a(2)*...*a(n-1) for n >= 1 and a(n) = f(n)-prevprime(f(n)-1) for n >= 2, where prevprime(x) is the largest prime < x.

2, 3, 5, 7, 11, 13, 17, 19, 31, 29, 23, 41, 43, 37, 89, 59, 53, 67, 79, 71, 137, 109, 239, 167, 199, 47, 83, 97, 61, 373, 101, 179, 193, 131, 151, 73, 263, 593, 139, 113, 157, 103, 241, 181, 397, 233, 617, 311, 191, 229, 271, 269, 127, 223, 331, 337, 211, 163

Offset

1,1

Comments

The terms are easily seen to be distinct. It is conjectured that every element is prime. Do all primes occur in the sequence?

First 1000 terms are primes. - Mauro Fiorentini, Aug 01 2020

From Carl R. White, Apr 15 2026: (Start)

Begin with f(1)=4. For each n, find the smallest prime a(n) such that there exists a prime p = f(n)-a(n); i.e., a(n) and p are a Goldbach sum for f(n). Set f(n+1)=f(n)*a(n) and repeat.

There appears to be no good reason that these p should always equal prevprime(f(n)-1), nor that these alternative a(n) should never repeat, nor it is proven that the main sequence contains only primes. Nonetheless, both methods correspond exactly for well over 100 terms.

Are these sequences the same? (End)

Links

Mauro Fiorentini, Table of n, a(n) for n = 1..1000

Mathematica

<<NumberTheory`PrimeQ` (* Load ProvablePrimeQ function, needed below. *)
f[1]=4; f[n_] := f[n]=f[n-1]a[n-1]; a[n_] := a[n]=Module[{i}, For[i=2, True, i++, If[ProvablePrimeQ[f[n]-i], Return[i]]]]

Prog

(MuPAD) f := 4:for n from 1 to 50 do a := f-numlib::prevprime(f-2):f := f*a:print(a) end_for

Crossrefs

Cf. A068193 has the indices of the primes in this sequence. A066631 has the sequence of f's. Also see A067836.

Sequence in context: A177000 A117843 A293667 * A225083 A002200 A181561

Adjacent sequences: A068189 A068190 A068191 * A068193 A068194 A068195

Keyword

nonn

Author

Frank Buss (fb(AT)frank-buss.de), Feb 19 2002

Extensions

Edited by Dean Hickerson, Jun 10 2002

Status

approved