

A235623


Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal.


0




OFFSET

1,3


COMMENTS

Number n is in the sequence, if and only if pi(n) = 2*pi(n/2), where pi(x) is the number of primes<=x. Indeed, all primes from interval (n/2, n] appear in prime power factorization of n! with exponent 1, while all primes from interval (0, n/2] appear in n! with exponents >1. However, it follows from Ehrhart's link that, for n>=22, pi(n) < 2*pi(n/2). Therefore, a(9)=21 is the last term of the sequence.
m is in this sequence if and only if the number of prime divisors of [m/2]! equals the number of unitary prime divisors of m!  Peter Luschny, Apr 29 2014


LINKS

Table of n, a(n) for n=1..10.
Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271274.


EXAMPLE

21! = 2^20*3^9*5^4*7^3*11*13*17*19. Here 4 primes with exponent 1 and 4 primes with exponents >1, so 21 is in the sequence.


MAPLE

with(numtheory): a := proc(n) factorset(n!); factorset(iquo(n, 2)!);
`if`(nops(%% minus %) = nops(%), n, NULL) end: seq(a(n), n=0..30); # Peter Luschny, Apr 28 2014


PROG

(PARI) isok(n) = {f = factor(n!); sum(i=1, #f~, f[i, 2] == 1) == sum(i=1, #f~, f[i, 2] > 1); } \\ Michel Marcus, Apr 20 2014


CROSSREFS

Cf. A056171, A177329, A177333, A177334, A240537, A240588, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139, A241148, A241289.
Sequence in context: A310938 A289631 A076680 * A001074 A214206 A026316
Adjacent sequences: A235620 A235621 A235622 * A235624 A235625 A235626


KEYWORD

nonn,fini,full


AUTHOR

Vladimir Shevelev, Apr 20 2014


STATUS

approved



