A340511 -id:A340511 - cp's OEIS frontend

A349930 a(n) is the number of groups of order A340511(n) which have no subgroup of order d, for some divisor d of A340511(n).

1, 1, 3, 2, 1, 2, 7, 1, 1, 2, 3

Offset: 1

Bernard Schott, Dec 05 2021

Also, number of NCLT groups of order A340511(n); NCLT means "Non-Converse Lagrange Theorem" because the converse to Lagrange's theorem does not hold for the groups of this sequence.

All terms up to a(11) come from Curran's link.

			A340511(1) = 12, and there is only one group of order 12: Alt(4) = A_4 which has no subgroup of order d = 6, despite the fact that 6 divides 12, hence a(1) = 1.
A340511(3) = 36, and there are 3 such NCLT groups of order 36: one group (C_3)^2 X C_4 has no subgroup of order 12, and the two groups A_4 X C_3 and (C_2)^2 X C_9 have no subgroup of order 18, hence a(3) = 3.

M. J. Curran, Non-CLT groups of small order, Comm. Algebra 11 (1983), 111-126.
Des MacHale and J. Manning, Converse Lagrange Theorem Orders and Supersolvable Orders, Journal of Integer Sequences, 2016, Vol. 19, #16.8.7.
Index entries for sequences related to groups.

Cf. A000001, A340511, A341048.

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

Original entry on oeis.org

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

A066085 Orders of non-supersolvable groups.

Original entry on oeis.org

12, 24, 36, 48, 56, 60, 72, 75, 80, 84, 96, 108, 112, 120, 132, 144, 150, 156, 160, 168, 180, 192, 196, 200, 204, 216, 224, 225, 228, 240, 252, 264, 276, 280, 288, 294, 300, 312, 320, 324, 336, 348, 351, 360, 363, 372, 375, 384, 392, 396, 400, 405, 408, 420

Offset: 1

Reiner Martin, Dec 29 2001

nonn

A finite group is supersolvable if it has a normal series with cyclic factors. Huppert showed that a finite group is supersolvable iff the index of any maximal subgroup is prime.

All multiples of non-supersolvable orders are non-supersolvable orders. - Des MacHale, Dec 22 2003

			a(1)=12 is in the sequence since the alternating group on 4 elements is the smallest group which is not supersolvable.

B. Huppert, Über das Produkt von paarweise vertauschbaren zyklischen Gruppen, Math. Z. 58 (1954).
Des MacHale and J. Manning, Converse Lagrange Theorem Orders and Supersolvable Orders, Journal of Integer Sequences, 2016, Vol. 19, #16.8.7.

Cf. A000001, A066083, A340511.

For primitive terms see A340517.

More terms from Des MacHale, Dec 22 2003

A340517 Primitive orders of non-supersolvable groups.

Original entry on oeis.org

12, 56, 75, 80, 196, 200, 294, 351, 363, 405

Offset: 1

Des MacHale, Jan 24 2021

Primitive terms of A066085.

A closely related sequence would be the primitive terms in A340511, but for that we need more terms of A340511.

Cf. A066085, A340511.

A341048 Numbers m such that there is a group of order m that is not supersolvable (NSS) but "converse Lagrange theorem" (CLT).

Original entry on oeis.org

224, 2464, 2912, 3159, 3808, 4256, 5152, 6318, 6496, 8288, 9184, 9632

Offset: 1

Bernard Schott, Feb 04 2021

The converse to Lagrange's theorem does not hold. A340511 lists the numbers n such that there exists a group of order n which has no subgroup of order d, for some divisor d of n; they are called "non-converse Lagrange theorem" (NCLT) orders.
A finite group is supersolvable (SS) if it has a normal series of subgroups with cyclic factors; A066085 lists the numbers for which there exists a group of order n that is not supersolvable; they are called a "non-supersolvable" (NSS) order.
Theorem: Every NCLT order is an NSS order (see MacHale-Manning, 2016, Theorem 3, page 2); hence A340511 is a subsequence of A066085.
However, there exist infinitely many NSS orders that are not NCLT orders (see MacHale-Manning, 2016, Corollary 19, page 6) and these NSS-CLT orders are listed in this sequence.
Theorem: The number 224*p is an NSS-CLT order for all primes p <> 2, 3, 5, 7, 31 (see MacHale-Manning, 2016, Theorem 18, page 6). So, 10528, 11872, 13216, 13664, 15008, 15904, ... are other terms.

			There exist 197 groups of order 224, and one of these groups is NSS-CLT (see MacHale-Manning, 2016, Theorem 8, page 5); this group is NSS (A066085) but satisfies the converse of Lagrange theorem (CLT): for all divisors d of 224, this group has at least one subgroup of order d; hence, 224 is a term.

Des MacHale and J. Manning, Converse Lagrange Theorem Orders and Supersolvable Orders, Journal of Integer Sequences, 2016, Vol. 19, #16.8.7.

Equals A066085 \ A340511.

Cf. A340517.

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A349930 a(n) is the number of groups of order A340511(n) which have no subgroup of order d, for some divisor d of A340511(n).

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

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A066085 Orders of non-supersolvable groups.

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A340517 Primitive orders of non-supersolvable groups.

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A341048 Numbers m such that there is a group of order m that is not supersolvable (NSS) but "converse Lagrange theorem" (CLT).

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