A007503 -id:A007503 - cp's OEIS frontend

A309731 Expansion of Sum_{k>=1} k * x^k/(1 - x^k)^3.

1, 5, 9, 20, 20, 48, 35, 76, 72, 110, 77, 204, 104, 196, 210, 288, 170, 405, 209, 480, 378, 440, 299, 816, 425, 598, 594, 868, 464, 1200, 527, 1104, 858, 986, 910, 1800, 740, 1216, 1170, 1960, 902, 2184, 989, 1980, 1890, 1748, 1175, 3216, 1470, 2475, 1938, 2704, 1484, 3456, 2090

Offset: 1

Ilya Gutkovskiy, Aug 14 2019

nonn

Dirichlet convolution of natural numbers (A000027) with triangular numbers (A000217).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A000027, A000203, A000217, A007437, A007503, A034715, A038040, A064987, A309732.

with(numtheory): seq(n*(tau(n)+sigma(n))/2, n=1..30); # Ridouane Oudra, Nov 28 2019

nmax = 55; CoefficientList[Series[Sum[k x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j, j (j + 1)/2, j, n], {n, 1, 55}]
Table[n (DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 55}]

a(n)=sumdiv(n,d,binomial(n/d+1,2)*d); \\ Andrew Howroyd, Aug 14 2019

a(n)=n*(numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+1, 2)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Apr 19 2021

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k/(1 - x^k)^2.
a(n) = n * (d(n) + sigma(n))/2.
Dirichlet g.f.: zeta(s-1) * (zeta(s-2) + zeta(s-1))/2.
a(n) = Sum_{k=1..n} k*tau(gcd(n,k)). - Ridouane Oudra, Nov 28 2019

A362683 Expansion of Sum_{k>0} (1/(1 - k*x^k)^2 - 1).

Original entry on oeis.org

2, 7, 10, 25, 16, 78, 22, 153, 136, 298, 34, 1254, 40, 1214, 2004, 3825, 52, 11385, 58, 20894, 18932, 25006, 70, 150002, 18826, 115274, 199828, 389510, 88, 1334624, 94, 1725281, 2131188, 2360266, 725948, 14878299, 112, 10486958, 22329428, 37317986, 124, 120957336, 130

Offset: 1

Seiichi Manyama, Jun 13 2023

nonn

Cf. A338662, A363639, A363640.

Cf. A007503, A055225, A359103, A363646.

a[n_] := DivisorSum[n, (n/#)^# * (# + 1) &]; Array[a, 50] (* Amiram Eldar, Jul 17 2023 *)

PARI
```
a(n) = sumdiv(n, d, (n/d)^d*(d+1));
```

a(n) = Sum_{d|n} (n/d)^d * (d+1) = A055225(n) + A359103(n).

If p is prime, a(p) = 1 + 3*p.

A363646 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^2 - 1).

Original entry on oeis.org

2, 11, 58, 565, 6256, 95762, 1647094, 33752329, 774919720, 20029303030, 570623341234, 17838801038274, 605750213184520, 22226048320465666, 875787902918124708, 36894332593824661521, 1654480523772673528372, 78693266840741507386757

Offset: 1

Seiichi Manyama, Jun 13 2023

nonn

Cf. A023887, A338663, A363647, A363648.

Cf. A007503, A362683.

a[n_] := DivisorSum[n, (n/#)^n * (# + 1) &]; Array[a, 20] (* Amiram Eldar, Jul 17 2023 *)

PARI
```
a(n) = sumdiv(n, d, (n/d)^n*(d+1));
```

a(n) = Sum_{d|n} (n/d)^n * (d+1).

A060710 Number of subgroups of dihedral group with 2n elements, counting conjugate subgroups only once, i.e., conjugacy classes of subgroups of the dihedral group.

Original entry on oeis.org

2, 5, 4, 8, 4, 10, 4, 11, 6, 10, 4, 16, 4, 10, 8, 14, 4, 15, 4, 16, 8, 10, 4, 22, 6, 10, 8, 16, 4, 20, 4, 17, 8, 10, 8, 24, 4, 10, 8, 22, 4, 20, 4, 16, 12, 10, 4, 28, 6, 15, 8, 16, 4, 20, 8, 22, 8, 10, 4, 32, 4, 10, 12, 20, 8, 20, 4, 16, 8, 20, 4, 33, 4, 10, 12, 16, 8, 20, 4, 28, 10, 10, 4

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

nonn

The total number of subgroups, counting conjugate subgroups as distinct, is A007503.

Also the number of subgroups of the group C_n x C_2 (where C_n is the cyclic group with n elements).

			The dihedral group D6 is isomorphic to the symmetric group S_3 and the conjugacy classes of subgroups are: the trivial group, the whole group, subgroup of order 2 generated by a transposition and the subgroup A_3 generated by the 3-cycles. So a(3) = 4.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001622, A007503, A062011.

A row of A216624.

a[n_] := DivisorSum[n, 3-Mod[#,2]&];
Array[a, 100] (* Jean-François Alcover, Jun 03 2019 *)

a(n)=if(n<1, 0, sumdiv(n,d, 3-d%2)) /* Michael Somos, Sep 20 2005 */

{ for (n=1, 1000, write("b060710.txt", n, " ", sumdiv(n, d, 3 - d%2)); ) } \\ Harry J. Smith, Jul 10 2009

def A060710(n): return add(3 - int(is_odd(d)) for d in divisors(n))
[A060710(n) for n in (1..83)] # Peter Luschny, Sep 12 2012

For even n, a(n) = 2*tau(n) + tau(n/2).
For odd n, a(n) = tau(2n) = 2*tau(n) = 2*A000005(n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001
From Michael Somos, Sep 20 2005: (Start)
Moebius transform is period 2 sequence [2, 3, ...].
G.f.: Sum_{k>0} x^k*(2+3x^k)/(1-x^(2k)) = Sum_{k>0} 2*x^(2k-1)/(1-x^(2k-1)) + 3*x^(2k)/(1-x^(2k)). (End)
a(n) = 4*tau(n) - tau(2n). - Ridouane Oudra, Jan 16 2023
Sum_{k=1..n} a(k) ~ n*(5*log(n) + 10*gamma - log(2) - 5)/2, where gamma is Euler's constant (A001622). - Amiram Eldar, Jan 21 2023

More terms from Vladeta Jovovic, Jul 15 2001

A363628 Expansion of Sum_{k>0} (1/(1-x^k)^3 - 1).

Original entry on oeis.org

3, 9, 13, 24, 24, 47, 39, 69, 68, 96, 81, 153, 108, 165, 170, 222, 174, 292, 213, 342, 302, 363, 303, 523, 375, 492, 474, 615, 468, 766, 531, 783, 686, 810, 726, 1101, 744, 999, 938, 1248, 906, 1402, 993, 1413, 1306, 1437, 1179, 1901, 1314, 1773, 1562, 1938, 1488, 2238, 1698

Offset: 1

Seiichi Manyama, Jun 12 2023

Cf. A007503, A116963.

Cf. A007437, A069153, A363610.

a[n_] := DivisorSum[n, Binomial[# + 2, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 05 2023 *)

PARI
```
a(n) = sumdiv(n, d, binomial(d+2, 2));
```

G.f.: Sum_{k>0} binomial(k+2,2) * x^k/(1 - x^k).

a(n) = Sum_{d|n} binomial(d+2,2).

A257644 First differences of A264100.

Original entry on oeis.org

1, 3, 8, 14, 24, 32, 48, 58, 77, 93, 115, 129, 163, 179, 207, 235, 271, 291, 336, 358, 406, 442, 482, 508, 576, 610, 656, 700, 762, 794, 874, 908, 977, 1029, 1087, 1139, 1239, 1279, 1343, 1403, 1501, 1545, 1649, 1695, 1785, 1869, 1945, 1995, 2129, 2189

Offset: 0

Franklin T. Adams-Watters, Nov 05 2015

nonn

Cumulative sum of A007503, starting with 1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000005, A000203, A007503, A264100.

a257644 n = a257644_list !! n
a257644_list = scanl (+) 1 a007503_list -- Reinhard Zumkeller, Nov 09 2015

alist(n)=my(x);vector(n,k,if(k==1,x=1,x+=sigma(k-1)+numdiv(k-1)))

from math import isqrt
def A257644(n): return (-(s:=isqrt(n))*(s*(s+4)+5)+sum(((q:=n//w)+1)*(q+(w<<1)+4) for w in range(1,s+1))>>1)+1 # Chai Wah Wu, Oct 31 2023

A364063 Expansion of Sum_{k>0} k * x^k / (1 - x^(2*k-1)).

Original entry on oeis.org

1, 3, 4, 5, 8, 7, 8, 14, 10, 11, 18, 13, 17, 22, 16, 17, 26, 26, 20, 30, 22, 23, 42, 25, 30, 38, 28, 38, 42, 31, 32, 55, 44, 35, 50, 37, 38, 65, 50, 41, 63, 43, 56, 62, 46, 58, 66, 62, 50, 81, 52, 53, 100, 55, 56, 78, 58, 74, 94, 74, 68, 86, 80, 65, 90, 67, 82, 124, 70, 71, 98, 86, 92, 117, 76, 77

Offset: 1

Seiichi Manyama, Jul 04 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000203, A007503, A064216, A336840.

Cf. A364066, A364085.

a[n_] := DivisorSum[2*n - 1, # + 1 &]/2; Array[a, 100] (* Amiram Eldar, Jul 04 2023*)

PARI
```
a(n) = sumdiv(2*n-1, d, d+1)/2;
```

a(n) = (1/2) * Sum_{d | 2*n-1} (d+1) = A007503(2*n-1)/2.
G.f.: Sum_{k>0} x^k / (1 - x^(2*k-1))^2.
a(n) = A336840(A064216(n)). - Antti Karttunen, Nov 30 2024

A037852 Number of normal subgroups of dihedral group with 2n elements.

Original entry on oeis.org

2, 5, 3, 6, 3, 7, 3, 7, 4, 7, 3, 9, 3, 7, 5, 8, 3, 9, 3, 9, 5, 7, 3, 11, 4, 7, 5, 9, 3, 11, 3, 9, 5, 7, 5, 12, 3, 7, 5, 11, 3, 11, 3, 9, 7, 7, 3, 13, 4, 9, 5, 9, 3, 11, 5, 11, 5, 7, 3, 15, 3, 7, 7, 10, 5, 11, 3, 9, 5, 11, 3, 15, 3, 7, 7

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 04 2001

When n is an odd prime a(n) = 3.

Write D_{2n} as , then the subgroups are of the form for d|n or for d|n and 0 <= r < d. The normal subgroups are for d|n and for d|gcd(n,2) and 0 <= r < d. There are d(n) normal subgroups of the first type and sigma(gcd(n,2)) normal subgroups of the second type. - Jianing Song, Jul 21 2022

			a(4) = 6 since D_8 = <a, x | a^4 = x^2 = 1, x*a*x = a^(-1)> has 6 normal subgroups: {e}, {e,a^2}, {e,a,a^2,a^3}, {e,a^2,x,a^2*x}, {e,a^2,a*x,a^3*x} and D_8. The 4 subgroups {e,x}, {e,a*x}, {e,a^2*x} and {e,a^3*x} are not normal. - _Jianing Song_, Jul 21 2022

Antti Karttunen, Table of n, a(n) for n = 1..1001
Keith Conrad, Dihedral Groups II
The Group Properties Wiki, Subgroup structure of dihedral groups
Index entries for sequences related to groups

Cf. A062249, A007503, A062553.

Cf. A000005, A176040.

a(n) = numdiv(n) + 2 + (-1)^n \\ Michel Marcus, Jul 30 2013

a(n) = d(n) + 2 + (-1)^n. - Paul Boddington, Feb 02 2004

a(n) = A000005(n) + A176040(n). - Michel Marcus, Aug 19 2015

More terms from Michel Marcus, Jul 30 2013

A083573 Maximal number of subgroups in a non-Abelian group with n elements, or zero if there are no non-Abelian groups of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 10, 0, 8, 0, 16, 0, 10, 0, 35, 0, 28, 0, 22, 10, 14, 0, 54, 0, 16, 19, 28, 0, 28, 0, 158, 0, 20, 0, 78, 0, 22, 16, 76, 0, 36, 0, 40, 0, 26, 0, 236, 0, 64, 0, 46, 0, 212, 14, 98, 22, 32, 0, 80, 0, 34, 36, 937, 0, 52, 0, 58, 0, 52, 0, 272

Offset: 1

Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003

nonn

A group G is non-Abelian iff there are two elements x,y such that xy != yx. Then and are nontrivial subgroups whose order divides the order of G which therefore cannot be prime (neither the square of a prime: there are only two nonisomorphic groups of that order which are both abelian; see A051532 for more). This also implies that a(n) >= 2+2+2 = 6 for all nonzero elements of this sequence and for even n=2m>4 there is the non-Abelian dihedral group D_m with A007503(m)=sigma(m)+tau(m)=A000005(m)+A000203(m), providing a lower bound. - M. F. Hasler, Dec 03 2007

			a(6)=6 because the only non-Abelian group with 6 elements is S_3 with 6 subgroups.

Eric M. Schmidt, Table of n, a(n) for n = 1..511

Cf. A018216, A061034.

Cf. A051532, A060689, A007503.

A083573 := function(n) local max, grp, i; max := 0; for i in [1..NumberSmallGroups(n)] do grp := SmallGroup(n, i); if (not IsAbelian(grp)) then max := Maximum(max, Sum(ConjugacyClassesSubgroups(grp), Size)); fi; od; return max; end; # Eric M. Schmidt, Sep 07 2012

a(n) = 0 <=> A060689(n)=0 <=> n is in A051532 ; otherwise a(n) >= 6 and a(2n) >= A007503(n). - M. F. Hasler, Dec 03 2007

More terms from Eric M. Schmidt, Sep 07 2012

A105331 Numbers of the form 2^n*(2^(n+1)+2n+1) where 2^(n+1)+2n+1 is prime.

Original entry on oeis.org

3, 14, 52, 184, 656, 34688, 2118656, 134438912, 537346048, 9007202811510784, 2417851639318318791262208, 633825300114170432793740312576, 2535301200456572518883997515776

Offset: 1

Farideh Firoozbakht, Apr 28 2005

This sequence is a subsequence of A083874 (see A083874 & A105330).

This is because these numbers satisfy tau(n) + sigma(n) = 2n when n = 2^k * p with p is prime; for instance tau(14) + sigma(14) = 4 + 24 = 28 = 2 x 14. [See References.] - Bernard Schott, Apr 07 2017

			9007202811510784 is in the sequence because 9007202811510784 = 2^26*(2^27 + 2*26 + 1) and 2^27 + 2*26 + 1 is prime.

J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, Problème 723, page 93.

Cf. A007503, A083874, A105330.

Do[If[PrimeQ[2^(m + 1) + 2m + 1], Print[2^m(2^(m + 1) + 2m + 1)]], {m, 0, 110}]
2^# (2^(#+1)+2#+1)&/@Select[Range[0,100],PrimeQ[2^(#+1)+2#+1]&] (* Harvey P. Dale, Nov 13 2012 *)

a(n) = 2^A105330(n)*(2^(A105330(n)+1) + 2*A105330(n) + 1). - Bernard Schott, Apr 07 2017

cp's OEIS Frontend

A309731 Expansion of Sum_{k>=1} k * x^k/(1 - x^k)^3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A362683 Expansion of Sum_{k>0} (1/(1 - k*x^k)^2 - 1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A363646 Expansion of Sum_{k>0} (1/(1 - (k*x)^k)^2 - 1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A060710 Number of subgroups of dihedral group with 2n elements, counting conjugate subgroups only once, i.e., conjugacy classes of subgroups of the dihedral group.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

Formula

Extensions

A363628 Expansion of Sum_{k>0} (1/(1-x^k)^3 - 1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A257644 First differences of A264100.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

PARI

Python

A364063 Expansion of Sum_{k>0} k * x^k / (1 - x^(2*k-1)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A037852 Number of normal subgroups of dihedral group with 2n elements.

Views