A240751 -id:A240751 - cp's OEIS frontend

A241289 Numbers n for which in the factorization of n! over distinct terms of A050376, the numbers of prime and nonprime terms are equal.

7, 8, 9, 13, 18, 22, 37, 57, 71

Offset: 1

Vladimir Shevelev, Apr 18 2014

a(10), if it exists, should be more than 5000. Is a(9)=71 the last term of sequence? - Peter J. C. Moses, Apr 19 2014

One can prove that a(9)=71 indeed is the last term of this sequence. - Vladimir Shevelev, Apr 19 2014.

			7 is in the sequence, since 7! in the considered factorization is 5*7*9*16, and here we have 2 primes and 2 nonprimes.

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

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. A177329, A177333, A177334, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764, A240905, A240906, A241123, A241124, A241139, A241148.

Terms a(7) - a(9) from Peter J. C. Moses, Apr 19 2014

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

Original entry on oeis.org

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

A306772 a(n) is the least number k such that k! is divisible by (k+1)^n but not by (k+1)^(n+1).

Original entry on oeis.org

1, 5, 14, 17, 11, 31, 23, 35, 39, 44, 47, 99, 83, 59, 153, 164, 71, 95, 79, 125, 89, 134, 285, 199, 311, 263, 167, 119, 296, 188, 159, 329, 543, 209, 143, 223, 299, 384, 395, 323, 251, 679, 349, 179, 279, 747, 571, 485, 399, 404, 314, 527, 319, 335, 449, 511, 287, 239, 714

Offset: 0

Jinyuan Wang, Mar 09 2019

nonn

k+1 is not a prime.
a(n) + 1 is 17-smooth in DATA. - David A. Corneth, Mar 15 2019
But fails at n 99, 114, 125, 127, 130, 135, 143, 146, ... - Michel Marcus, Apr 30 2019

			For n = 1, 1! = 1 is not divisible by 2, 2! = 2 is not divisible by 3, 3! = 6 is not divisible by 4, 4! = 24 is not divisible by 5, and 5! = 120 is divisible by 6 but not 36. Therefore a(1) = 5. - _Michael B. Porter_, Apr 21 2019

Cf. A061768, A082482, A133481, A240751.

Array[Block[{k = 1}, While[Nand[Mod[k!, (k + 1)^#] == 0, Mod[k!, (k + 1)^(# + 1)] != 0], k++]; k] &, 58] (* Michael De Vlieger, Mar 11 2019 *)

a(n) = {my(k=1); while((k! % (k+1)^n) || !(k! % (k+1)^(n+1)), k++); k; }

a(n) = A133481(n+1) - 1.
a(n) >= A061768(n).
If n = floor((p^j-1)/(j*(p-1)))-1, a(n) <= p^j-1 for prime p. For example, (p = 2), a(n) <= 2^j-1 for n = floor((2^j-1)/j)-1 (A082482(j)-1).

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A241289 Numbers n for which in the factorization of n! over distinct terms of A050376, the numbers of prime and nonprime terms are equal.

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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.

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A306772 a(n) is the least number k such that k! is divisible by (k+1)^n but not by (k+1)^(n+1).

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