A177329 -id:A177329 - cp's OEIS frontend

A235623 Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal.

0, 1, 4, 7, 8, 9, 13, 19, 20, 21

Offset: 1

Vladimir Shevelev, Apr 20 2014

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

			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.

Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271-274.

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.

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

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

A241270 Numbers with the property that in their factorization over distinct terms of A050376, the sums of prime and nonprime terms of A050376 are equal.

Original entry on oeis.org

126, 468, 624, 792, 880, 1056, 1150, 2900, 3264, 4606, 5824, 6375, 6624, 8320, 9856, 10388, 11375, 12798, 13650, 16400, 16704, 19250, 20925, 30135, 32625, 36720, 39150, 39900, 53784, 56446, 56925, 57000, 59500, 63455, 65520, 71400, 71500, 72471

Offset: 1

Vladimir Shevelev, Apr 18 2014

nonn

The corresponding sequence of the sum over the primes, which equals the sum over the nonprimes, is 9, 13, 16, 13, 16, 16, 25, 29, 20, 49, 20, 25, 25, 20, 20, 53, 25, 81, 25, 41, 29, 25, 34, 49, 34, 25, 34, 29, 85, 169, 34, 29, 29, 49, 25, 29, 29, 49, ... - Wolfdieter Lang, Apr 25 2014

			126 and 468 are in the sequence since the factorizations are 2*7*9 and 4*9*13 respectively, and 2+7=9, 4+9=13.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A187039, A187042, A177329, A177333, A177334.

More terms from Peter J. C. Moses, Apr 18 2014

New extension from Wolfdieter Lang, Apr 25 2014

cp's OEIS Frontend

A235623 Numbers n for which in the prime power factorization of n!, the numbers of exponents 1 and >1 are equal.

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