A007503 -id:A007503 - cp's OEIS frontend

A363695 Expansion of Sum_{k>0} (1/(1-x^k)^5 - 1).

5, 20, 40, 90, 131, 265, 335, 585, 755, 1147, 1370, 2155, 2385, 3410, 4042, 5430, 5990, 8295, 8860, 11843, 13020, 16335, 17555, 23125, 23882, 29805, 32220, 39440, 40925, 51644, 52365, 64335, 67450, 79820, 82712, 101575, 101275, 120805, 125830, 148089, 149000, 179490

Offset: 1

Seiichi Manyama, Jun 16 2023

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

Cf. A007503, A116963, A363628, A363696.

Cf. A073570, A363605, A363608.

a[n_] := DivisorSum[n, Binomial[# + 4, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 05 2023 *)

PARI
```
a(n) = sumdiv(n, d, binomial(d+4, 4));
```

G.f.: Sum_{k>0} binomial(k+4,4) * x^k/(1 - x^k).

a(n) = Sum_{d|n} binomial(d+4,4).

A363696 Expansion of Sum_{k>0} (1/(1-x^k)^6 - 1).

Original entry on oeis.org

6, 27, 62, 153, 258, 545, 798, 1440, 2064, 3282, 4374, 6859, 8574, 12447, 15818, 21789, 26340, 36196, 42510, 56538, 66634, 85125, 98286, 126901, 142764, 178506, 203440, 249909, 278262, 343936, 376998, 457686, 506372, 602118, 659058, 791908, 850674, 1005129, 1094638

Offset: 1

Seiichi Manyama, Jun 16 2023

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

Cf. A007503, A116963, A363628, A363695.

Cf. A101289, A363606.

a[n_] := DivisorSum[n, Binomial[# + 5, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 05 2023 *)

PARI
```
a(n) = sumdiv(n, d, binomial(d+5, 5));
```

G.f.: Sum_{k>0} binomial(k+5,5) * x^k/(1 - x^k).

a(n) = Sum_{d|n} binomial(d+5,5).

A386438 a(n) = sigma(n) + omega(n) - n * Sum_{p|n, p prime} 1 / p.

Original entry on oeis.org

1, 3, 4, 6, 6, 9, 8, 12, 11, 13, 12, 20, 14, 17, 18, 24, 18, 26, 20, 30, 24, 25, 24, 42, 27, 29, 32, 40, 30, 44, 32, 48, 36, 37, 38, 63, 38, 41, 42, 64, 42, 58, 44, 60, 56, 49, 48, 86, 51, 60, 54, 70, 54, 77, 58, 86, 60, 61, 60, 109, 62, 65, 76, 96, 68, 86, 68, 90, 72, 88, 72, 137, 74, 77, 86, 100, 80, 100, 80, 132, 95, 85, 84, 145, 88, 89, 90, 130, 90, 144

Offset: 1

Wesley Ivan Hurt, Jul 21 2025

nonn

For each divisor d of n, add 1 if n/d is prime, else add d.

Cf. A000010 (phi), A000203 (sigma), A001221 (omega), A005171, A007503, A010051, A069359, A348219.

Table[Sum[d^(1 - PrimePi[n/d] + PrimePi[n/d - 1]), {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} d^c(n/d), where c = A005171.
a(n) = Sum_{d|n} (d + c(d) - phi(d)*omega(n/d)), where c = A010051.
a(n) = A000203(n) + A001221(n) - A069359(n).
a(n) = A007503(n) - A348219(n).

A062553 Number of Abelian subgroups of the dihedral group with 2n elements.

Original entry on oeis.org

2, 5, 5, 9, 7, 13, 9, 16, 12, 19, 13, 24, 15, 25, 19, 29, 19, 33, 21, 36, 25, 37, 25, 44, 28, 43, 31, 48, 31, 53, 33, 54, 37, 55, 39, 63, 39, 61, 43, 68, 43, 71, 45, 72, 51, 73, 49, 82, 52, 81, 55, 84, 55, 89, 59, 92, 61, 91, 61, 102, 63, 97, 69, 103, 69, 107, 69, 108, 73

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001

nonn

The rank of the fundamental group with genus one of the D_2n; cobordism category in dimension 1+1, with D_2n the dihedral group of order 2n. - C. Segovia, Dec 05 2012

C. Segovia, The classifying space of the 1+1 dimensional G-cobordism category, arXiv:1211.2144 [math.AT], 2012-2013.

Cf. A062249, A007503.

a[n_] := If[OddQ[n], n, 3n/2] + DivisorSigma[0, n];
Array[a, 69] (* Jean-François Alcover, Feb 24 2019 *)

a(n) = n + tau(n) if n is odd, (3/2)*n + tau(n) if n is even, where tau(n) = the number of divisors of n (A000005).

Formula and more terms from Vladeta Jovovic, Jul 05 2001

A221856 Square numbers n for which sigma(n) + d(n) is also a perfect square.

Original entry on oeis.org

9, 16, 36, 278784, 251381025, 2390623236, 240055902025, 354328515025, 1022960302225, 1298266542225, 6824670333649, 9433221536025, 16614933604164, 33314541015625, 474897452948164, 500279020818724, 1387202986290276, 2162188899431649, 16053088159411524

Offset: 1

Jayanta Basu, Apr 10 2013

nonn

			16 is in the list since 16 = 4^2 and sigma(16)+d(16) = 36 = 6^2. Also 278784 = 528^2 and sigma(278784)+d(278784) = 883600 = 940^2.

Cf. A007503.

Sqd[n_] := Sqrt[DivisorSigma[1, n] + DivisorSigma[0, n]]; t = {}; Do[If[IntegerQ[Sqd[n^2]], AppendTo[t, n^2]], {n, 1500000}];t

a(11)-a(19) from Donovan Johnson, Apr 10 2013

A224441 Numbers n such that sigma(n)+d(n) and sigma(n+1)+d(n+1) are perfect squares.

Original entry on oeis.org

61, 2369, 4469, 8460, 13208, 44790, 162734, 281560, 283938, 334469, 500465, 533045, 609953, 871853, 962247, 1317885, 1741445, 1792745, 2499845, 3013246, 4099031, 5646629, 7009389, 7012135, 8396781, 8740480, 8822093, 11146381, 11957693, 15002679, 18895694

Offset: 1

Jayanta Basu, Apr 09 2013

nonn

			61 is in the list since sigma(61)+d(61)=64 and sigma(62)+d(62)=100.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000005, A000203, A007503.

Sqd[n_] := Sqrt[DivisorSigma[1, n] + DivisorSigma[0, n]]; t = {}; Do[If[IntegerQ[Sqd[n]] && IntegerQ[Sqd[n + 1]], AppendTo[t, n]], {n, 20000000}]; t
SequencePosition[Table[If[IntegerQ[Sqrt[DivisorSigma[0,n]+DivisorSigma[1,n]]],1,0],{n,189*10^5}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 31 2020 *)

is(n)=issquare(sigma(n)+numdiv(n))&&issquare(sigma(n+1)+ numdiv(n+1)) \\ Charles R Greathouse IV, Apr 09 2013

A264100 Sum of the lengths of the arithmetic progressions in {1,2,3,...,n}, including trivial arithmetic progressions of lengths 1 and 2.

Original entry on oeis.org

0, 1, 4, 12, 26, 50, 82, 130, 188, 265, 358, 473, 602, 765, 944, 1151, 1386, 1657, 1948, 2284, 2642, 3048, 3490, 3972, 4480, 5056, 5666, 6322, 7022, 7784, 8578, 9452, 10360, 11337, 12366, 13453, 14592, 15831, 17110, 18453, 19856, 21357, 22902, 24551

Offset: 0

Gionata Neri, Nov 03 2015

nonn

Conjecture: the second differences give A007503(n+1), the sum of the divisors (A000203) plus the number of divisors (A000005) of n+1.

The first differences trivially are the total length of such sequences that end in n+1. Mapping each sequence to a different sequence by adding 1 to each term, we see that the second differences are the number of sequences up to n+2 that include both 1 and n+2. For each divisor d of n+1, there is a single such sequence of length d+1 (with increment (n+1)/d). The second difference is then sum_{d|n+1} d+1, which is sigma(n+1) + tau(n+1), as claimed. - Franklin T. Adams-Watters, Nov 05 2015

			For n = 3 the arithmetic progressions are (1), (2), (3), (1, 2), (1, 3), (2, 3), (1, 2, 3) and the respective lengths are (1), (1), (1), (2), (2), (2), (3), so a(3) = 1 + 1 + 1 + 2 + 2 + 2 + 3 = 12.
The first difference at 2, sequences ending with 3, are (3), (1, 3), (2, 3), and (1, 2, 3), total length 8 = 12-4. The second difference at 2, sequences starting with 1 and ending with 4 are (1, 4) and (1, 2, 3, 4), total length 6 = 26 - 2*12 +4.

Cf. A000005, A000203, A007503, A257644.

vector(50, n, n--; n + sum(k=2, n, k*floor((n-1)/(k-1))*(2*n-(k-1)*floor((n+k-2)/(k-1)))/2)) \\ Altug Alkan, Nov 04 2015

a(n) = n + Sum_{k=2..n} k*floor((n-1)/(k-1))*(2*n-(k-1)*floor((n+k-2)/(k-1)))/2.

A163164 Positions n such that A163163(n) is not prime.

Original entry on oeis.org

1, 4, 6, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94, 96, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2009

			a(1)=1 because A163163(1)=1 is not prime. a(2)=4 because A163163(4)=6 is not prime.

Cf. A000005, A000203, A007503, A163163.

33 removed by R. J. Mathar, Jul 25 2009

A195864 Numbers k such that sigma(k) + tau(k) is a perfect number (A000396).

Original entry on oeis.org

3, 14, 15, 168, 326, 5414, 33357341, 4324809536

Offset: 1

Jeff Kruse, Oct 26 2011

A002093(848) < a(9) <= 1155321467378283328. - Donovan Johnson, Nov 03 2011

Cf. A000396, A002093, A007503 (sigma(n) + tau(n)).

perfect = {6, 28, 496, 8128, 33550336, 8589869056, 137438691328}; Select[Range[10000], MemberQ[perfect, DivisorSigma[0, #] + DivisorSigma[1, #]] &] (* T. D. Noe, Nov 03 2011 *)

a(8) from Donovan Johnson, Nov 03 2011

A248671 Number of subgroups of the dihedral group Dn that are intersections of some maximal subgroups.

Original entry on oeis.org

1, 4, 5, 4, 7, 15, 9, 4, 5, 21, 13, 15, 15, 27, 27, 4, 19, 15, 21, 21, 35, 39, 25, 15, 7, 45, 5, 27, 31, 79, 33, 4, 51, 57, 51, 15, 39, 63, 59, 21, 43, 103, 45, 39, 27, 75, 49, 15, 9, 21, 75, 45, 55, 15, 75, 27, 83, 93, 61, 79, 63, 99, 35, 4, 87, 151, 69, 57, 99, 151

Offset: 1

Nandor Sieben, Oct 11 2014

nonn

Maximal subgroups are counted.
Smallest such subgroup is the Frattini subgroup.
These subgroups are called intersection subgroups in Ernst and Sieben link.

Dana C. Ernst, Nandor Sieben, Impartial achievement and avoidance games for generating finite groups, arXiv:1407.0784 [math.CO], 2014.

Cf. A007503.

for n in [1..22] do
  G:=DihedralGroup(2*n);
  Ge:=Elements(G);
  mse:=List(MaximalSubgroups(G),s->List(s,el->Position(Ge,el)));
  C:=Combinations(mse);
  Remove(C,1); # empty intersection is removed
  I:=List(C,Intersection);
  Sort(I);
  I:=Unique(I);
  Print(Size(I),",");
od;

a[n_] := With[{f = FactorInteger[n][[All, 1]]}, Sum[d+1, {d, Divisors[Times @@ f]}]-1];
Array[a, 70] (* Jean-François Alcover, Aug 29 2018, after Andrew Howroyd *)

a(n) = my(f=factor(n)[,1]); sumdiv(prod(i=1, #f, f[i]), d, d+1 ) - 1; \\ Andrew Howroyd, Jul 02 2018

a(n) = A007503(n) - 1 for squarefree n. - Andrew Howroyd, Jul 02 2018

a(23)-a(70) from Andrew Howroyd, Jul 02 2018

cp's OEIS Frontend

A363695 Expansion of Sum_{k>0} (1/(1-x^k)^5 - 1).

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A363696 Expansion of Sum_{k>0} (1/(1-x^k)^6 - 1).

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A386438 a(n) = sigma(n) + omega(n) - n * Sum_{p|n, p prime} 1 / p.

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Programs

Mathematica

Formula

A062553 Number of Abelian subgroups of the dihedral group with 2n elements.

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Keywords

Comments

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Programs

Mathematica

Formula

Extensions

A221856 Square numbers n for which sigma(n) + d(n) is also a perfect square.

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Examples

Crossrefs

Programs

Mathematica

Extensions

A224441 Numbers n such that sigma(n)+d(n) and sigma(n+1)+d(n+1) are perfect squares.

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Mathematica

PARI

A264100 Sum of the lengths of the arithmetic progressions in {1,2,3,...,n}, including trivial arithmetic progressions of lengths 1 and 2.

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Comments

Examples

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Programs

PARI

Formula

A163164 Positions n such that A163163(n) is not prime.

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Examples

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Extensions

A195864 Numbers k such that sigma(k) + tau(k) is a perfect number (A000396).

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