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A235623 Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal. 0
0, 1, 4, 7, 8, 9, 13, 19, 20, 21 (list; graph; refs; listen; history; text; internal format)
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. 271-274.

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

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Last modified March 6 04:14 EST 2021. Contains 341841 sequences. (Running on oeis4.)