A219953 a(1) = 1; for n > 1, a(n) = smallest integer > a(n-1) such that a(n)*a(i)+1 is semiprime for all 1 <= i <= n-1.

1, 3, 8, 38, 86, 318, 504, 3600, 8132, 83160, 116850, 202272, 399126, 6190086, 8756916, 25253676, 309709400, 1112878446, 1478724036, 11062089360, 97331025386

Offset

1,2

Comments

This is to A034881 as semiprimes A001358 are to primes A000040.

a(20) > 6*10^9. - Giovanni Resta, Jul 26 2015

Links

Table of n, a(n) for n=1..21.

Example

a(1) = 1 by definition.
a(2) = 3: 3 > 1, and 1*3 + 1 = 4 = 2^2 is semiprime.
a(3) = 8: 8 > 3, and 1*8 + 1 = 9 = 3^2 is semiprime, and 3*8 + 1 = 25 = 5^2 is semiprime.
a(4) = 38: 38 > 8, and 1*38 + 1 = 39 = 3*13 is semiprime, and 3*38 + 1 = 115 = 5*23 is semiprime, and 8*38 + 1 = 305 = 5*61 is semiprime.
From Michel Marcus, Jul 26 2015: (Start)
The resulting semiprimes are:
    4;
    9,  25;
   39, 115,  305;
   87, 259,  689,  3269;
  319, 955, 2545, 12085, 27349;
  ...
(End)

Maple

A219953 := proc(n)
    option remember;
    if n= 1 then
        1;
    else
        for a from procname(n-1)+1 do
            issp := true ;
            for i from 1 to n-1 do
                if numtheory[bigomega]( a*procname(n-i)+1) = 2 then
                    ;
                else
                    issp := false;
                    break ;
                end if;
            end do:
            if issp then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Dec 15 2012

Mathematica

a = {1}; Do[k = a[[n - 1]] + 1; While[! AllTrue[(k a[[n - #]] + 1) & /@ Range@ (n - 1), Total[Last /@ FactorInteger@ #] == 2 &], k++]; AppendTo[a, k], {n, 2, 13}]; a (* Michael De Vlieger, Jul 26 2015, Version 10 *)

Prog

(PARI) ok(v, n, k) = {v[n] = k; for (j=1, n-1, if (bigomega(1+v[n]*v[j]) != 2, return (0)); ); return (1); }
lista(nn) = {print1(k=1, ", "); v = [k]; for (n=2, nn, k = v[n-1]+1; v = concat(v, k); while (! ok(v, n, k), k++); v[n] = k; print1(k, ", "); ); } \\ Michel Marcus, Jul 26 2015

Crossrefs

Cf. A030063, A001358, A034881, A219761.

Sequence in context: A147523 A123981 A123985 * A286075 A089066 A099030

Adjacent sequences: A219950 A219951 A219952 * A219954 A219955 A219956

Keyword

nonn,more

Author

Jonathan Vos Post, Dec 01 2012

Extensions

a(14)-a(17) from Luke March, Jul 26 2015

a(18)-a(19) from Giovanni Resta, Jul 26 2015

a(20)-a(21) from Tyler Busby, Jan 31 2023

Status

approved