%I #21 Nov 07 2023 03:15:42

%S 3,19,39,81,165,333,335,673,1347,1349,1351,1353,1355,1357,1359,2721,

%T 2723,2725,2727,5457,5459,5461,5463,5465,5467,5469,10941,10943,10945,

%U 10947,21897,21899,21901,21903,21905,21907,21909,43821,43823,43825,43827,43829,43831

%N a(1)=3; for n > 1, a(n) = 1 + a(n-1) + gcd( a(n-1)*(a(n-1)+2), A073829(a(n-1)) ).

%C The first differences are 16, 20, 42, etc. They are either 2 or in A075369 or in A008864, see A167054.

%C A proof follows from Clement's criterion of twin primes.

%D E. Trost, Primzahlen, Birkhäuser-Verlag, 1953, pages 30-31.

%H Amiram Eldar, <a href="/A167053/b167053.txt">Table of n, a(n) for n = 1..206</a>

%H P. A. Clement, <a href="http://www.jstor.org/stable/2305816">Congruences for sets of primes</a>, Amer. Math. Monthly, 56 (1949), 23-25.

%e a(2) = 1 + 3 + gcd(3*5, 4*(2! + 1) + 3) = 19.

%p A073829 := proc(n) n+4*((n-1)!+1) ; end proc:

%p A167053 := proc(n) option remember ; local aprev; if n = 1 then 3; else aprev := procname(n-1) ; 1+aprev+gcd(aprev*(aprev+2),A073829(aprev)) ; end if; end proc:

%p seq(A167053(n),n=1..60) ; # _R. J. Mathar_, Dec 17 2009

%t A073829[n_] := 4((n-1)! + 1) + n;

%t a[1] = 3;

%t a[n_] := a[n] = 1 + a[n-1] + GCD[a[n-1] (a[n-1] + 2), A073829[a[n-1]]];

%t Array[a, 60] (* _Jean-François Alcover_, Mar 25 2020 *)

%Y Cf. A073829, A008864, A167054.

%Y Cf. A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Oct 27 2009

%E Definition shortened and values from a(4) on replaced by _R. J. Mathar_, Dec 17 2009