A241786 Smallest k such that the number of the first even exponents in prime power factorization of (2*k)! is n, or a(n)=0 if there is no such k.

1, 6, 3, 5, 10, 24, 27, 169, 924, 3168, 720, 3208, 408, 35421, 50878, 73920, 18757, 204513, 134418, 295680, 427684, 2746710, 6867848, 14476645, 7278558, 3668406, 737564, 245340483, 1931850660, 1514239096, 3228582476, 1325085081, 16188866895, 33517640073

Offset

0,2

Comments

Conjecture: 1) All a(n)>0; 2) a(2*n+1)>a(2*n).

Conjecture (2) is wrong because a(24) = 7278558 >= a(25) = 3668406.

a(35) > 10^11; a(36) = 8036409193. - Hiroaki Yamanouchi, Sep 29 2014

References

P. Erdős, P. L. Graham, Old and new problems and results in combinatorial number theory, L'Enseignement Mathematique, Imprimerie Kunding, Geneva, 1980.

Links

Giovanni Resta, Table of n, a(n) for n = 0..44 (terms a(0)-a(34) and a(36) from Hiroaki Yamanouchi)

D. Berend, Parity of exponents in the factorization of n!, J. Number Theory, 64 (1997), 13-19.

Example

a(2)=3, since (2*3)!= 2^4*3^2*5, and here the number of the first even exponents is 2.

Prog

(PARI) nbev(n) = {f = factor(n); nbe = 0; i = 1; while ((i <= #f~) && ((f[i, 2] % 2) == 0), i++; nbe++); nbe; }
a(n) = {k = 0; while(nbev((2*k)!) != n, k++); k; } \\ Michel Marcus, Apr 30 2014

Crossrefs

Cf. A240537, A240606, A240620.

Sequence in context: A085653 A022462 A319894 * A019151 A367976 A143506

Adjacent sequences: A241783 A241784 A241785 * A241787 A241788 A241789

Keyword

nonn

Author

Vladimir Shevelev, Apr 28 2014

Extensions

More terms from Peter J. C. Moses, May 06 2014

a(21)-a(33) from Hiroaki Yamanouchi, Sep 29 2014

Status

approved