A167184 -id:A167184 - cp's OEIS frontend

A167185 Largest prime power <= n that is not prime.

1, 1, 1, 4, 4, 4, 4, 8, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 27, 27, 27, 27, 27, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 1

Michael B. Porter, Oct 29 2009

			For a(14), 10, 12, and 14 are not prime powers, and 11 and 13 are prime powers but they are prime. Since 9 = 3^3 is a prime power, a(14) = 9.

Robert Price, Table of n, a(n) for n = 1..2000

List of nonprime prime powers: A025475.
Next nonprime prime power: A167184.
Previous prime power including primes: A031218.

Array[SelectFirst[Range[#, 1, -1], Or[And[! PrimeQ@ #, PrimePowerQ@ #], # == 1] &] &, 74] (* Michael De Vlieger, Jun 14 2017 *)

isA025475(n) = (omega(n) == 1 & !isprime(n)) || (n == 1)
A167185(n) = {local(m);m=n;while(!isA025475(m),m--);m}

from sympy import factorint
def A167185(n): return next(filter(lambda m:len(f:=factorint(m))<=1 and max(f.values(),default=2)>1, range(n,0,-1))) # Chai Wah Wu, Oct 25 2024

p = [n for n in (1..81) if (is_prime_power(n) or n == 1) and not is_prime(n)]
r = [[p[i]]*(p[i+1] - p[i]) for i in (0..9)]
print([y for x in r for y in x]) # Peter Luschny, Jun 14 2017

A211391 The number of divisors d of n! such that d < A000793(n) (Landau's function g(n)) and the symmetric group S_n contains no elements of order d.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 2, 2, 6, 4, 15, 15, 24, 29, 33, 63, 55, 126, 117, 110, 103, 225, 212, 288, 282, 319, 428, 504, 774, 859, 943, 924, 1336, 1307, 1681, 1869, 2097, 2067, 2866, 3342, 3487, 5612, 5567, 5513, 5549, 9287, 9220, 11594, 11524, 11481, 11403, 18690

Offset: 1

Alexander Gruber, Feb 07 2013

nonn

This sequence gives the number of divisors d of |S_n| such that d < Lambda(n) (where Lambda(n) = the largest order of an element in S_n) for which S_n contains no element of order d. These divisors constitute a set of 'missing' element orders of S_n.

For computational purposes, the smallest divisor d0(n) of n! = |S_n| for which S_n has no element of order d0(n) is the smallest divisor of n! which is not the least common multiple of an integer partition of n. Thus d0(n) is given by the smallest prime power >= n+1 that is not prime (with the exception of n = 3 and 4, for which d0(n) = 6).

			For n = 7, we refer to the following table:
Symmetric Group on 7 letters.
  # of elements of order  1 ->    1
  # of elements of order  2 ->  231
  # of elements of order  3 ->  350
  # of elements of order  4 ->  840
  # of elements of order  5 ->  504
  # of elements of order  6 -> 1470
  # of elements of order  7 ->  720
  # of elements of order  8 ->    0
  # of elements of order  9 ->    0
  # of elements of order 10 ->  504
  # of elements of order 12 ->  420
  (All other divisors of 7! -> 0.)
So there are two missing element orders in S_7, whence a(7) = 2.

d0(n) is equal to A167184(n) for n >= 5.

Cf. A000793 (Landau's function g(n)), A057731, A211392.

for n in [1..25] do
D := Set(Divisors(Factorial(n)));
O := { LCM(s) : s in Partitions(n) };
L := Max(O);
N := D diff O;
#{ n : n in N | n lt L };
end for;

More terms from Alois P. Heinz, Feb 11 2013

cp's OEIS Frontend

A167185 Largest prime power <= n that is not prime.

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