A204142 a(1)=1 and for n>1, a(n) equals the smallest prime greater than a(n-1) of the form a(i)*a(j)+2 for some i, j < n.

1, 3, 5, 7, 11, 13, 17, 19, 23, 37, 41, 43, 53, 59, 61, 67, 71, 73, 79, 97, 113, 131, 163, 179, 181, 211, 223, 239, 241, 293, 307, 337, 367, 373, 397, 409, 439, 487, 491, 499, 631, 673, 691, 719, 733, 739, 769, 853, 881, 883, 907, 919, 991

Offset: 1

Author

Max Alekseyev, Jan 16 2012

Keywords

• nonn

Comments

Is this sequence finite?

For n>1, a(n)-2 is prime or semiprime.

If a(n) is the lesser of a twin prime pair, then a(n+1) = a(n)+2 (= a(1)*a(n)+2). - M. F. Hasler, Feb 11 2012

There are exactly 3478486 terms < 10^18 with a(3478486) = 999999462621744733. - Terentyev Oleg, Mar 09 2012

Examples

			a(2) = a(1)^2+2, a(3) = a(1)*a(2)+2, a(4) = a(1)*a(3)+2, a(5) = a(2)^2+2, a(6)=a(1)*a(5)+2, a(7) = a(1)*a(5)+2, a(8) = a(2)*a(3)+2, ...
a(16027) = 14228123711 is the lesser of a twin prime pair, it is followed by the greater twin prime, a(16028) = 14228123713.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..20000
• Discussion at forum dxdy.ru (in Russian)

Programs

Mathematica

a = {1}; s = {}; While[s = Union[s, Select[a*a[[-1]] + 2, PrimeQ]]; s != {} && Length[a] < 100, AppendTo[a, s[[1]]]; s = Rest[s]]; a (* T. D. Noe, Mar 12 2012 *)

PARI

a=Set(p=1); for(n=1,999, until(bigomega(p-2)<3 & !setminus(Set(factor(p-2)[,1]~),a), p=nextprime(p+2)); a=setunion(a,Set(p))); vecsort(eval(a)) \\ M. F. Hasler, Feb 10 2012

PARI

{a=[r=1]; for(n=1, 9999, while(a[r]^2+2<=a[n], r++); m=0; s=r; for(i=r, n, while(s>1&&a[s-1]*a[i]+2>a[n], s--);  for( j=s, i, m & a[j]*a[i]+2>m & break; isprime(a[j]*a[i]+2) & m=a[j]*a[i]+2 ));  a=concat(a, m))} \\ M. F. Hasler, Feb 10 2012