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A212848 Least prime factor of n-th central trinomial coefficient (A002426).

%I A212848 #20 Jun 20 2017 23:13:43
%S A212848 1,1,3,7,19,3,3,3,3,43,7,3,113,73,3,3,3,3,3,3,3,3,3,3,3,3,3,7,17,3,
%T A212848 719,7,3,3,3,3,967,9539,3,17,47,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%U A212848 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,19
%N A212848 Least prime factor of n-th central trinomial coefficient (A002426).
%C A212848 A002426(n) is prime for n = 2, 3, 4, no more through 10^5. A002426 is semiprime iff A102445(n) = 2 (as is the case for n = 5, 6, 7, 9, 10, 12, 13).
%H A212848 Robert Israel, <a href="/A212848/b212848.txt">Table of n, a(n) for n = 0..729</a>
%F A212848 a(n) = A020639(A002426(n)).
%e A212848 a(9) = 43 because A002426(9) = 3139 = 43 * 73.
%p A212848 A002426:= gfun:-rectoproc({(n+2)*a(n+2)-(2*n+3)*a(n+1)-3*(n+1)*a(n) = 0, a(0)=1, a(1)=1},a(n),remember):
%p A212848 lpf:= proc(n) local F;
%p A212848     F:= map(proc(t) if t[1]::integer then t[1] else NULL fi end proc,
%p A212848        ifactors(n, easy)[2]);
%p A212848     if nops(F) > 0 then min(F)
%p A212848     else min(numtheory:-factorset(n))
%p A212848     fi
%p A212848 end proc:
%p A212848 lpf(1):= 1:
%p A212848 map(lpf @ A002426, [$0..100]); # _Robert Israel_, Jun 20 2017
%t A212848 a = b = 1; t = Join[{a, b}, Table[c = ((2 n - 1) b + 3 (n - 1) a)/n; a = b; b = c; c, {n, 2, 100}]]; Table[FactorInteger[n][[1, 1]], {n, t}] (* _T. D. Noe_, May 30 2012 *)
%o A212848 (PARI) a(n) = my(x=polcoeff((1 + x + x^2)^n, n)); if (x==1, 1, vecmin(factor(x)[,1])); \\ _Michel Marcus_, Jun 20 2017
%Y A212848 Cf. A000040, A002426, A020639, A102445, A212791.
%K A212848 nonn,easy
%O A212848 0,3
%A A212848 _Jonathan Vos Post_, May 28 2012