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a(n) is the smallest nonnegative x such that the Euler polynomial x^2 + x + 41 has exactly n distinct prime proper divisors.
9

%I #29 Dec 24 2018 03:26:18

%S 0,41,420,2911,38913,707864,6618260,78776990,725005500

%N a(n) is the smallest nonnegative x such that the Euler polynomial x^2 + x + 41 has exactly n distinct prime proper divisors.

%C The Euler polynomial gives primes for consecutive x from 0 to 39.

%C For numbers x for which x^2 + x + 41 is not prime, see A007634.

%C For composite numbers of the form x^2 + x + 41, see A145292.

%e a(1)=0 because when x=0 then x^2+x+41=41 (1 distinct prime divisor);

%e a(2)=41 because when x=41 then x^2+x+41=1763=41*43 (2 distinct prime divisors);

%e a(3)=420 because when x=420 then x^2+x+41=176861=47*53*71 (3 distinct prime divisors);

%e a(4)=2911 because when x=2911 then x^2+x+41=8476873=41*47*53*83 (4 distinct prime divisors);

%e a(5)=38913 because when x=38913 then x^2+x+41=1514260523=43*47*61*71*173 (5 distinct prime divisors);

%e a(6)=707864 because when x=707864 then x^2+x+41=501072150401=41*43*47*53*71*1607 (6 distinct prime divisors);

%e a(7)=6618260 because when x=6618260 then x^2+x+41=43801372045901=41*43*47*61*83*131*797 (7 distinct prime divisors);

%e a(8)=78776990 because when x=78776990 then x^2+x+41=6205814232237131=41*43*61*71*97*131*167*383 (8 distinct prime divisors).

%e a(9)=725005500: a(9)^2 + a(9) + 41 = 525632975755255541 = 41*43*47*53*61*71*151*397*461. - _Hugo Pfoertner_, Mar 05 2018

%t a = {}; Do[x = 1; While[Length[FactorInteger[x^2 + x + 41]] < k - 1, x++ ]; AppendTo[a, x]; Print[x], {k, 2, 10}]; a

%Y Cf. A005846, A007634, A097822, A145292, A145294, A145295.

%Y Cf. A228122. - _Zak Seidov_, Feb 03 2016

%K nonn,more

%O 1,2

%A _Artur Jasinski_, Oct 07 2008

%E Corrected and edited, a(8) added by _Zak Seidov_, Jan 31 2016

%E Example for a(8) corrected by _Hugo Pfoertner_, Mar 02 2018

%E a(9) from _Hugo Pfoertner_, Mar 05 2018