A167053 - OEIS

A167053

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

3, 19, 39, 81, 165, 333, 335, 673, 1347, 1349, 1351, 1353, 1355, 1357, 1359, 2721, 2723, 2725, 2727, 5457, 5459, 5461, 5463, 5465, 5467, 5469, 10941, 10943, 10945, 10947, 21897, 21899, 21901, 21903, 21905, 21907, 21909, 43821, 43823, 43825, 43827, 43829, 43831

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OFFSET

1,1

COMMENTS

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

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

REFERENCES

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

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..206

P. A. Clement, Congruences for sets of primes, Amer. Math. Monthly, 56 (1949), 23-25.

EXAMPLE

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

MAPLE

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

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:

seq(A167053(n), n=1..60) ; # R. J. Mathar, Dec 17 2009

MATHEMATICA

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

a[1] = 3;

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

Array[a, 60] (* Jean-François Alcover, Mar 25 2020 *)

CROSSREFS

Cf. A073829, A008864, A167054.

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

Sequence in context: A147237 A117674 A114704 * A131542 A227878 A042371

Adjacent sequences: A167050 A167051 A167052 * A167054 A167055 A167056

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Oct 27 2009

EXTENSIONS

Definition shortened and values from a(4) on replaced by R. J. Mathar, Dec 17 2009

STATUS

approved