A120275 Smallest prime factor of the odd Catalan number A038003(n).

5, 3, 3, 7, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset

2,1

Comments

A038003(n) = binomial(2^(n+1)-2, 2^n-1)/(2^n).

a(n) <> 3 iff the base-3 representation of 2^n-1 has no 2's. Conjecture: this only occurs for n = 2, 5, 8. I verified it up to n = 10^4. - Robert Israel, Nov 18 2015

Links

Table of n, a(n) for n=2..86.

Example

a(2) = 5 because A038003(2) = 5.
a(3) = 3 because A038003(3) = 429 = 3*11*13.

Maple

f:= proc(n) local m;
  m:= 2^n-1;
  if has(convert(m, base, 3), 2) then return 3 fi;
  min(numtheory:-factorset(binomial(2*m, m)/(m+1)));
end proc:
seq(f(n), n=2..1000); # Robert Israel, Nov 18 2015

Mathematica

f[n_] := Block[{p = 2, m = Binomial[2^(n+1)-2, 2^n-1]/(2^n)}, While[Mod[m, p] > 0, p = NextPrime@ p]; p]; Array[f, 27, 2] (* Robert G. Wilson v, Nov 14 2015 *)

Crossrefs

Cf. A038003, A000108.

Sequence in context: A056597 A329973 A019624 * A021656 A337571 A244683

Adjacent sequences: A120272 A120273 A120274 * A120276 A120277 A120278

Keyword

nonn

Author

Alexander Adamchuk, Jul 04 2006

Extensions

a(16)-a(28) from Robert G. Wilson v, Nov 14 2015

a(29)-a(86) from Robert Israel, Nov 18 2015

Status

approved