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

Author

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

Keywords

• nonn

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.

Crossrefs

Cf. A115627, A240620.

Programs

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

Extensions

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

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