A118887 -id:A118887 - cp's OEIS frontend

A024619 Numbers that are not powers of primes p^k (k >= 0); complement of A000961.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Clark Kimberling

The sequence of numbers divisible by a prime number of primes coincides with this up to 210, which has 4 prime factors. - Lior Manor, Aug 23 2001
A085970(n) = Max{k: a(k)<=n}.
Numbers n such that LCM of proper divisors of n equals neither 1 nor n. - Labos Elemer, Dec 01 2004
a(n) provides bases b in which automorphic numbers m^2 ending with m in base b exist. In the complement there aren't any automorphic numbers. - Martin Renner, Dec 07 2011
Numbers with at least 2 distinct prime factors. - Jonathan Sondow, Oct 17 2013
There exists an equiangular n-gon whose edge lengths form a permutation of 1, 2, ..., n if and only if n is in the sequence (see Woeginger's survey and Munteanu & Munteanu). - Jonathan Sondow, Oct 17 2013
Numbers that are the product of two relatively prime factors. These numbers are used in testing a sequence for multiplicativity. - Michael Somos, Jun 02 2015
A theorem from Donald McCarthy: Let d be any positive integer which is not a prime power; then there exists a finite group whose order is divisible by d but which contains no subgroup of order d (see link and A340511). - Bernard Schott, Dec 04 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 8719 terms from Daniel Forgues)
Donald McCarthy, Sylow's theorem is a sharp partial converse to Lagrange's theorem, Mathematische Zeitschrift, 113, 383-384 (1970).
Marius Munteanu and Laura Munteanu, Rational equiangular polygons, Applied Math., 4 (2013), 1460-1465.
Laurentiu Panaitopol, Some of the properties of the sequence of powers of prime numbers, Rocky Mountain Journal of Mathematics, Volume 31, Number 4, Winter 2001.
Eric Weisstein's World of Mathematics, Prime Power
Wikipedia, Prime power
G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.
Günter Ziegler and Brady Haran, Cannons and Sparrows, Numberphile video (2018).

Cf. A000040, A000961 (complement), A001221, A014963, A020500, A085970.
Cf. A340511.
Subsequence of A080257.

a024619 n = a024619_list !! (n-1)
a024619_list = filter ((== 0) . a010055) [1..]
-- Reinhard Zumkeller, Nov 17 2011

IsA024619:=func< n | not IsPrime(n) and not (t and IsPrime(b) where t, b, A024619(n)%20%5D;%20//%20_Klaus%20Brockhaus">:=IsPower(n)) >; [ n: n in [2..200] | IsA024619(n) ]; // _Klaus Brockhaus, Feb 25 2011

a := proc(n) numtheory[factorset](n); if 1 < nops(%) then n else NULL fi end:
seq(a(i), i=1..110); # Peter Luschny, Aug 11 2009

Select[Range@111, Length@FactorInteger@# > 1 &] (* Robert G. Wilson v, Dec 07 2005 *)

is(n)=n>5 && !isprimepower(n) \\ Charles R Greathouse IV, Mar 21 2013

from sympy import primepi
from sympy.ntheory.primetest import integer_nthroot
def A024619(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

def A024619_list(n) :
    return [k for k in (2..n) if not k.is_prime() and not k.is_prime_power()]
A024619_list(112)  # Peter Luschny, Feb 03 2012 [corrected by Terry D. Grant, Sep 16 2020]

A001221(a(n)) > 1.
A014963(a(n)) = 1.
A020500(a(n)) = 1. - Benoit Cloitre, Aug 26 2003
A010055(a(n)) = 0. - Reinhard Zumkeller, Nov 17 2011
a(n) ~ n. - Charles R Greathouse IV, Mar 21 2013
a(n) ~ n - pi(n) [See Panaitopol]. - N. J. A. Sloane, Sep 27 2020
A118887(a(n)) > 0. - Jonathan Sondow, Oct 17 2013

A118888 Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk such that the remaining imbalance is minimized.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 3, 24, 1, 732, 1, 720, 48, 144, 2

Offset: 1

Hugo Pfoertner, May 26 2006

The position of weight 1 is kept fixed at position 1. Mirror configurations are counted only once. For n not a prime power, the sequence equals A118887.

			a(5)=1: The configuration minimizing the remaining imbalance with respect to the center of the circle is [1 4 3 2 5] (and its mirror image).
Examples of minimum imbalance configurations not in A118887:
a(7)=1: [1 4 7 2 3 5 6];
a(8)=2: [1 4 7 3 6 2 5 8], [1 7 4 3 6 5 2 8];
a(9)=3: [1 5 9 2 7 3 4 8 6], [1 5 9 4 2 6 7 3 8], [1 6 5 4 9 2 3 7 8];
a(11)=1: [1 8 9 5 2 6 10 7 3 4 11];
a(13)=1: [1 2 7 12 13 4 5 3 8 6 11 9 10];
a(16)=144: lexicographically earliest [1 3 5 13 16 7 10 2 14 4 6 9 12 8 11 15];
a(17)=2: [1 7 3 17 10 9 15 2 14 6 5 4 16 8 13 12 11],
[1 8 9 3 16 4 12 13 14 2 10 5 6 7 17 11 15] and their mirror configurations (e.g. [1 11 12 13 8 ...]) both produce a center of gravity with distance 2.1884*10^(-7) from the center of a circle with radius 1. All other configurations produce greater distances, e.g. [1 3 11 16 9 5 7 12 14 4 10 8 2 15 13 6 17] -> 2.5126*10^(-7). - _Hugo Pfoertner_, Oct 24 2019

G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.

Cf. A118887, A326921.

a(17) corrected by Hugo Pfoertner, Oct 24 2019

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A024619 Numbers that are not powers of primes p^k (k >= 0); complement of A000961.

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A326921 Triangle (1 <= k <= n) read by rows: row n gives lexicographically earliest evenly spaced arrangement of weights 1...n around the circumference of a circular disk minimizing the resulting imbalance.

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A118888 Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk such that the remaining imbalance is minimized.

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