A056866 -id:A056866 - cp's OEIS frontend

A096490 Numbers k such that sigma_2(k) >= (3/2) * k^2, where sigma_2(k) is the sum of the squares of the divisors of k.

60, 120, 168, 180, 240, 252, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 756, 780, 792, 840, 900, 924, 936, 960, 1008, 1020, 1080, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, 1512, 1560, 1584, 1620, 1680, 1740, 1764, 1800, 1848, 1860

Offset: 1

Labos Elemer, Jun 25 2004

From Amiram Eldar, Aug 16 2024: (Start)
All the terms are divisible by 6 because sigma_2(k)/k^2 < 3*zeta(2)/4 = 1.2337... < 3/2 for odd numbers k, and sigma_2(k)/k^2 < 8*zeta(2)/9 = 1.462... < 3/2 for numbers k that are not divisible by 3.
There are no 3-smooth numbers (A003586) in this sequence, but for any 5-rough number (A007310) k  > 1 there are infinitely many 3-smooth numbers m such that their product k*m is a term.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 25, 259, 2578, 25823, 258026, 2580715, 25806329, 258066116, 2580658731, ... . Apparently, the asymptotic density of this sequence exists and equals 0.025806... . (End)

			For k = 60: 1 + 4 + 9 + 16 + 25 + 36 + 100 + 144 + 225 + 400 + 900 + 3600 = 5460 > (3/2) * 3600 = 5400.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001157, A056866, A118671 (primitive terms).

Cf. A001221, A003586, A007310, A013661.

Do[s=DivisorSigma[2, n]/(n^2); If[Greater[s, 3/2], Print[n]], {n, 1, 10000}]
Select[Range[2000],DivisorSigma[2,#]/#^2>=3/2&] (* Harvey P. Dale, Mar 05 2013 *)

is(n)=sigma(n,-2) >= 3/2 \\ Charles R Greathouse IV, Feb 03 2018

A001221(a(n)) >= 3. - Amiram Eldar, Aug 16 2024

Name corrected by Charles R Greathouse IV, Feb 03 2018

A119648 Orders for which there is more than one simple group.

Original entry on oeis.org

20160, 4585351680, 228501000000000, 65784756654489600, 273457218604953600, 54025731402499584000, 3669292720793456064000, 122796979335906113871360, 6973279267500000000000000, 34426017123500213280276480

Offset: 1

N. J. A. Sloane, Jul 29 2006

nonn

All such orders are composite numbers (since there is only one group of any prime order).
Orders which are repeated in A109379.
Except for the first number, these are the orders of symplectic groups C_n(q)=Sp_{2n}(q), where n>2 and q is a power of an odd prime number (q=3,5,7,9,11,...). Also these are the orders of orthogonal groups B_n(q). - Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010
a(1) = 20160 = 8!/2 is the order of the alternating simple group A_8 that is isomorphic to the Lie group PSL_4(2), but 20160 is also the order of the Lie group PSL_3(4) that is not isomorphic to A_8 (see A137863). - Bernard Schott, May 18 2020

			From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010: (Start)
a(1)=|A_8|=8!/2=20160,
a(2)=|C_3(3)|=4585351680,
a(3)=|C_3(5)|=228501000000000, and
a(4)=|C_4(3)|=65784756654489600. (End)

See A001034 for references and other links.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites]. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory. See also. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
W. Kimmerle et al., Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups, Proc. London Math. Soc., (3) 60 (1990), 89-122. [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
David A. Madore, Orders of non-Abelian simple groups
Wikipedia, Classification of finite simple groups [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]
Index entries for sequences related to groups

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A001034 (orders of simple groups without repetition), A109379 (orders with repetition), A137863 (orders of simple groups which are non-cyclic and non-alternating).

sp(n, q) 1/2 q^n^2.(q^(2.i) - 1, i, 1, n) [From Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010] [This line contained some nonascii characters which were unreadable]

For n>1, a(n) is obtained as (1/2) q^(m^2)Prod(q^(2i)-1, i=1..m) for appropriate m>2 and q equal to a power of some odd prime number. [Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010]

Extended up to the 10th term by Dushan Pagon (dushanpag(AT)gmail.com), Jun 27 2010

A085736 Numbers n such that all groups of order n are solvable.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Andrew Niedermaier, Jul 20 2003

nonn

			The symmetric and alternating groups on 5 elements are not solvable and have orders 60 and 120 respectively.

Eric Weisstein's World of Mathematics, Solvable groups

See A056866, the complementary set of numbers, which is the main entry for this question.

A113171 Short legs 'A' of exactly 7 primitive Pythagorean triangles.

Original entry on oeis.org

660, 1092, 1140, 1155, 1260, 1320, 1365, 1380, 1428, 1540, 1560, 1740, 1785, 1820, 1860, 1980, 1995, 2184, 2220, 2340, 2380, 2415, 2436, 2460, 2508, 2580, 2604, 2660, 2805, 2820, 2856, 2860, 2940, 3003, 3036, 3060, 3108, 3120, 3135, 3180, 3192, 3220, 3300

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 25 2008

nonn

			Examples of triples: 660.779.1021, 660.989.1189, 660.2989.3061, 660.4331.4381, 660.12091.12109, 660.27221.27229, 660.108899.108901
1092.1325.1717, 1092.1595.1933, 1092.6035.6133, 1092.8245.8317, 1092.33115.33133, 1092.74525.74533, 1092.298115.298117

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A056866 Orders of non-solvable groups. A093006 Referring to the triangle in A093005, sequence contains the least term with maximal number of divisors. A138605 Short legs of more than 3 primitive Pythagorean triangles. A033993 Numbers that are divisible by exactly four different primes.

PythagoreanAs[a_]:=(q={};k=0;Do[y=(a^2+b^2)^0.5;c=IntegerPart[y];If[c==y,p=0;If[GCD[a,b,c]==1,AppendTo[q,a.b.c];k++ ]],{b,a+1,a^2}];PrependTo[q,k];q);lst={};Do[If[PythagoreanAs[n][[1]]==7,Print[n];AppendTo[lst,n]],{n,6*10^2,2*10^3}];lst

a^2+b^2=c^2

More terms from Ray Chandler, Jan 22 2020

A201733 Number of isomorphism classes of polycyclic groups (or solvable groups) of order n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1

Offset: 1

W. Edwin Clark, Dec 04 2011

nonn

For finite groups solvable is equivalent to polycyclic.

Muniru A Asiru, Table of n, a(n) for n = 1..500
Wikipedia, Polycyclic group
Wikipedia, Solvable group

a:=[];;
N:=120;;
for n in [1..N] do
a[n]:=0;;
for j in [1..NrSmallGroups(n)] do
   if IsPcGroup(SmallGroup(n,j)) = true then
    a[n]:=a[n]+1;
   fi;
  od;
  Print(a[n],",");
od;

a(n) = A000001(n) for n < 60.

a(n) <= A000001(n) with equality if and only if n is not in A056866. In particular a(n) = A000001(n) for odd n (this is the Feit-Thompson theorem). - Benoit Jubin, Mar 30 2012

A216480 Primitive non-solvable numbers: orders of non-solvable groups such that all groups with order a proper divisor of that order are solvable.

Original entry on oeis.org

60, 168, 504, 1092, 2448, 5616, 6072, 9828, 25308, 28224, 32736, 39732, 51888, 74412, 150348, 194472, 285852, 456288, 546312, 612468, 721392, 1024128, 1285608, 1934868, 2097024, 2165292, 2328648, 2588772, 3594432, 3822588, 5544672, 5848428, 6324552, 7174332, 8487168, 9095592

Offset: 1

Charles R Greathouse IV, Sep 11 2012

Primitive elements of A056866; consequently, each term is divisible by 4 and either 3 or 5.

That is, numbers n such that n is in A056866, but no smaller m dividing n is in A056866. - Charles R Greathouse IV, May 09 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A056866.

list(lim)={
    my(v=List([5616]),t);
    forprime(p=2,log(lim)\log(8)+2,
        listput(v,(4^p-1)<1 && p%5<4, listput(v,p^2\2*p))
    );
    vecsort(select(n->n<=lim,Vec(v)))
};

a(n) ~ kn^3 log^3 n, where k = 27/8. - Charles R Greathouse IV, Sep 11 2012

A352287 Numbers k such that, for every prime p dividing k, k has a nontrivial divisor which is congruent to 1 (mod p).

Original entry on oeis.org

1, 12, 24, 30, 36, 48, 56, 60, 72, 80, 90, 96, 105, 108, 112, 120, 132, 144, 150, 160, 168, 180, 192, 210, 216, 224, 240, 252, 264, 270, 280, 288, 300, 306, 315, 320, 324, 336, 351, 360, 380, 384, 392, 396, 400, 420, 432, 448, 450, 480, 495, 504, 520, 525, 528, 540, 546, 552, 560, 576, 600

Offset: 1

David Speyer, Mar 10 2022

nonn

When considering whether an integer k is the order of a finite simple group, the first thing one checks is whether the number of p-Sylow subgroups is forced to be 1 for some p dividing k. This occurs if the only divisor of k which is 1 (mod p) is 1 itself. This sequence consists of the numbers that survive this test.

			105 is in the sequence, since it is divisible by 7 which is 1 (mod 3), 21 which is 1 (mod 5), and 15 which is 1 (mod 7).

Peter Luschny, Table of n, a(n) for n = 1..10000
Mariano Suárez-Álvarez, On the density of the orders excluded by the Sylow theorems for simple groups, MathOverflow, 2021.
Index entries for sequences related to groups.

Cf. A001034, A056866, A060793, A338853.

divq[n_, p_] := AnyTrue[Rest @ Divisors[n], Mod[#, p] == 1 &]; q[1] = True; q[n_] := AllTrue[FactorInteger[n][[;; , 1]], divq[n, #] &]; Select[Range[600], q] (* Amiram Eldar, May 05 2022 *)

isok(k) = {my(f=factor(k), d=divisors(f)); for (i=1, #f~, if (vecsum(apply(x->((x % f[i,1]) == 1), d)) == 1, return(0)); ); return(1);} \\ Michel Marcus, Mar 11 2022

print([ n for n in range(1, 601)
        if set( prime_factors(n) )
        == set( p for p in prime_factors(n)
                for d in divisors(n)
                if d > 1 and d < n
                if p.divides(d - 1)
      ) ] )  # Peter Luschny, Mar 14 2022

cp's OEIS Frontend

A096490 Numbers k such that sigma_2(k) >= (3/2) * k^2, where sigma_2(k) is the sum of the squares of the divisors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A119648 Orders for which there is more than one simple group.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Other

Formula

Extensions

A085736 Numbers n such that all groups of order n are solvable.

Views

Author

Keywords

Examples

Links

Crossrefs

A113171 Short legs 'A' of exactly 7 primitive Pythagorean triangles.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A201733 Number of isomorphism classes of polycyclic groups (or solvable groups) of order n.

Views

Author

Keywords

Comments

Links

Programs

GAP

Formula

A216480 Primitive non-solvable numbers: orders of non-solvable groups such that all groups with order a proper divisor of that order are solvable.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A352287 Numbers k such that, for every prime p dividing k, k has a nontrivial divisor which is congruent to 1 (mod p).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage