A240537 Let a(n) be the least k such that in the prime power factorization of k! the exponents of primes p_1, ...,p_n are even, while the exponent of p_(n+1) is odd.

12, 6, 10, 20, 48, 54, 338, 875, 2849, 1440, 3841, 816, 59583, 101755, 40465, 37514, 409026, 268836, 591360, 855368, 5493420, 9627251, 28953290, 14557116, 7336812, 1475128, 127632241, 531296823, 3028478192, 2435868325, 1092228841, 32377733790, 472077979

Offset

1,1

Comments

The sequence is connected with a 1980-Erdős-Graham conjecture that, for every N, there exists an n such that in prime power factorization of n! at least N first exponents are even. In 1997, this conjecture was proved by D. Berend. A generalization was given by Y.-G. Chen (2003).

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 = 1..46 (first 36 terms from Hiroaki Yamanouchi)

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

Y.-G. Chen, On the parity of exponents in the standard factorization of n!, J. Number Theory, 100 (2003), 326-331.

Prog

(PARI) nbe(n) = {my(f = factor(n!)[, 2], nb = 0); for (i=1, #f, if (!(f[i] % 2), nb++, break); ); nb; }
a(n) = {my(i = 1); while (nbe(i) != n, i++); i; } \\ Michel Marcus, Nov 07 2018

Crossrefs

Cf. A115627, A240620.

Sequence in context: A070292 A283880 A084067 * A227354 A328043 A075247

Adjacent sequences: A240534 A240535 A240536 * A240538 A240539 A240540

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Apr 07 2014

Extensions

a(21)-a(30) from Giovanni Resta, Apr 07 2014

a(31)-a(33) from Hiroaki Yamanouchi, Sep 05 2014

Status

approved