A032741 -id:A032741 - cp's OEIS frontend

A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

0, 1, 6, 22, 56, 133, 252, 484, 798, 1344, 2002, 3157, 4368, 6441, 8630, 12112, 15504, 21274, 26334, 35014, 42762, 55133, 65780, 84349, 98336, 123124, 143304, 176373, 201376, 247380, 278256, 336744, 379000, 451402, 502250, 600055, 658008, 775733, 855042

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013664, A032741, A065608, A069153, A363604, A363605.

Cf. A000203, A001157, A001158, A001159, A001160.

a[n_] := DivisorSum[n, Binomial[# + 3, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^6)))

a(n) = my(f = factor(n)); (sigma(f, 5) + 5*sigma(f, 4) + 5*sigma(f, 3) - 5*sigma(f, 2) - 6*sigma(f)) / 120; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k+3,5) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+3,5).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_5(n) + 5*sigma_4(n) + 5*sigma_3(n) - 5*sigma_2(n) - 6*sigma_1(n)) / 120.
Dirichlet g.f.: zeta(s) * (zeta(s-5) + 5*zeta(s-4) + 5*zeta(s-3) - 5*zeta(s-2) - 6*zeta(s-1)) / 120.
Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

A054009 n read modulo (number of proper divisors of n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 1, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 1, 2, 0, 3, 0, 2, 0, 2, 0, 1, 2, 4, 0, 2, 0, 5, 0, 0, 0, 4, 0, 1, 0, 3, 1, 0, 0, 2, 0, 5, 1, 0, 0, 1, 0, 5, 0, 2, 3, 4, 2, 3, 0, 3, 0, 0, 0, 6, 0, 2, 0, 1, 2, 1, 0, 8, 1, 1, 0, 7, 1, 2, 0, 4, 0, 2, 1, 2, 0, 1, 2, 8, 0, 3, 4, 4, 0, 4, 0, 6, 0, 1

Offset: 2

Asher Auel, Jan 12 2000

nonn

Cf. A000005, A032741, A054008.

Maple
```
[ seq( i mod (tau(i) - 1), i=2..150) ];
```

Table[Mod[n,DivisorSigma[0,n]-1],{n,2,110}] (* Harvey P. Dale, Dec 05 2015 *)

a(n) = n % (numdiv(n) - 1); \\ Michel Marcus, Nov 21 2019

from sympy import divisor_count
def A054009(n): return n%(divisor_count(n)-1) # Chai Wah Wu, Mar 14 2023

a(n) = n mod (tau(n) - 1), for n>1.

A054015 a(n) is Chowla function of n read modulo (number of proper divisors of n), a(1) = 0 by convention.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 6, 0, 0, 2, 1, 0, 6, 0, 0, 1, 0, 0, 4, 0, 4, 2, 1, 0, 3, 1, 2, 2, 0, 0, 2, 1, 0, 1, 1, 0, 8, 0, 0, 0, 2, 0, 0, 0, 2, 2, 3, 0, 1, 0, 0, 3, 3, 0, 5, 0, 6, 3, 1, 0, 7, 1, 0, 2, 0, 0, 0, 2, 0, 1, 1, 0, 1, 0, 2, 1, 4, 0, 1, 0, 0, 2

Offset: 1

Asher Auel, Jan 17 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A032741, A048050, A054013, A054014.

with(numtheory): [seq((sigma(i)-i-1) mod (tau(i)-1),i=2..120);#i>1

A054015(n) = if(1==n,0,((sigma(n)-n-1) % (numdiv(n)-1))); \\ Antti Karttunen, Oct 20 2017

a(1) = 0; for n > 1, a(n) = A048050(n) mod A032741(n).

Description clarified by Antti Karttunen, Oct 20 2017

A063806 Numbers with a prime number of proper divisors.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104

Offset: 1

Henry Bottomley, Aug 20 2001

nonn

			15 has three proper divisors {1,3,5} and so is on the list; 16 has four {1,2,4,8} and so is not; 17 has one {1} and so is not; 18 has five {1,2,3,6,9} and so is.

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

A subset of A002808. Cf. A009087, A032741.

n=0; for (m=1, 10^9, if(isprime(numdiv(m) - 1), write("b063806.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Aug 31 2009

is(n)=isprime(numdiv(n)-1) \\ Charles R Greathouse IV, Sep 18 2015

A069930 Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 2, 3, 4, 4, 2, 6, 2, 4, 6, 4, 2, 7, 2, 6, 6, 4, 2, 8, 4, 4, 6, 6, 2, 10, 2, 5, 6, 4, 6, 10, 2, 4, 6, 8, 2, 10, 2, 6, 10, 4, 2, 10, 4, 7, 6, 6, 2, 10, 6, 8, 6, 4, 2, 14, 2, 4, 10, 6, 6, 10, 2, 6, 6, 10, 2, 13, 2, 4, 10, 6, 6, 10, 2, 10, 8, 4, 2, 14, 6, 4, 6, 8, 2, 16, 6, 6, 6, 4, 6

Offset: 1

Benoit Cloitre, May 05 2002

Number of r X s integer-sided rectangles such that r < s, r + s = 2n and r | s. - Wesley Ivan Hurt, Apr 24 2020

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A032741, A069283, A000005 (tau), A001227, A006218, A060831, A066743.

A066660(n) - 1.

Table[Count[Table[(n+k)/(n-k),{k,n-1}],?IntegerQ],{n,100}] (* _Harvey P. Dale, Jun 04 2019 *)

for(n=1,150,print1(sum(i=1,n-1,if((n+i)%(n-i),0,1)),","))

{a(n) = if( n<1, 0, numdiv(2*n) - 2)} /* Michael Somos, Aug 30 2012 */

a(n) = A032741(n) + A069283(n) = A000005(n) - 1 + A001227(n) - 1 = tau(n) + A001227(n) - 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002
Asymptotic formula: since sum(k=1, n, a(k)) = sum(k=1, n, tau(k)) + sum(k=1, n, A001227(k)) - 2*n = A006218(n) + A060831(n) - 2*n = 2*A006218(n) - A006218(floor(n/2)) - 2*n with A006218(0) = 0, A006218(n) = sum(k=1, n, tau(k)) and now, by Dirichlet's asymptotic expression A006218(n) = n*log(n) + n*(2*gamma-1) + O(n^theta) (gamma = 0.57721..; 1/4 <= theta < 1/2), we have sum(k=1, n, a(k)) = 2*n*log(n) - (n/2)*log(n) + o(n*log(n)) = 1.5*n*log(n) + o(n*log(n)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002
a(n) = tau(2*n) - 2. - Michael Somos, Aug 30 2012
Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 7), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019

A084113 Number of multiplications when calculating A084110(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 4, 2, 2, 2, 6, 1, 2, 2, 5, 1, 5, 1, 4, 4, 2, 1, 6, 2, 4, 2, 4, 1, 5, 2, 5, 2, 2, 1, 7, 1, 2, 4, 4, 2, 5, 1, 4, 2, 5, 1, 7, 1, 2, 4, 4, 2, 5, 1, 6, 3, 2, 1, 7, 2, 2, 2, 5, 1, 7, 2, 4, 2, 2, 2, 7, 1, 4, 4, 6, 1, 5, 1, 5, 5

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A000005(n)-1-A084114(n) = A032741(n)-A084114(n) = (A032741(n)+A084115(n))/2;

a(n) = 1 iff n is prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product.

Cf. A027750, A084110, A084114.

a084113 = f 0 1 . a027750_row where
   f c _ []     = c
   f c x (d:ds) = if r == 0 then f c x' ds else f (c + 1) (x * d) ds
                  where (x', r) = divMod x d
-- Reinhard Zumkeller, Jul 31 2014

A100577 Number of sets of divisors of n with an odd sum.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 8, 4, 8, 2, 32, 2, 8, 8, 16, 2, 32, 2, 32, 8, 8, 2, 128, 4, 8, 8, 32, 2, 128, 2, 32, 8, 8, 8, 256, 2, 8, 8, 128, 2, 128, 2, 32, 32, 8, 2, 512, 4, 32, 8, 32, 2, 128, 8, 128, 8, 8, 2, 2048, 2, 8, 32, 64, 8, 128, 2, 32, 8, 128, 2, 2048, 2, 8, 32, 32, 8, 128, 2, 512, 16, 8, 2

Offset: 1

Reinhard Zumkeller, Nov 29 2004

a(n) = A000079(A032741(n)).

Also number of subsets of divisors of n which do not contain 1; thus a(n) = (A100587(n)+1)/2. - Vladeta Jovovic, Jul 02 2007

			a(12) = #{{1}, {3}, {1,2}, {1,4}, {2,3}, {1,6}, {3,4}, {1,2,4}, {3,6}, {1,2,6}, {2,3,4}, {1,4,6}, {2,3,6}, {1,12}, {3,4,6}, {1,2,4,6}, {3,12}, {1,2,12}, {2,3,4,6}, {1,4,12}, {2,3,12}, {1,6,12}, {3,4,12}, {1,2,4,12}, {3,6,12}, {1,2,6,12}, {2,3,4,12}, {1,4,6,12}, {2,3,6,12}, {1,2,4,6,12}, {3,4,6,12}, {2,3,4,6,12}} = 32.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000079, A001227, A032741, A100587.

A100577 := proc(n)
    2^(numtheory[tau](n)-1) ;
end proc:
seq(A100577(n),n=1..100) ; # R. J. Mathar, Nov 10 2017

Table[2^(DivisorSigma[0, n] - 1), {n, 1, 100}] (* Jean-François Alcover, Feb 13 2018 *)

a(n)=2^(numdiv(n)-1) \\ Charles R Greathouse IV, Jan 19 2017

a(n) = 2^(A000005(n)-1).

A157928 a(n) = 0 if n < 2, = 1 otherwise.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Jaroslav Krizek, Mar 09 2009

A characteristic function which indicates whether n has a prime factorization n = product p_i^e_i where p_i are primes (A000040) and e_i nonnegative exponents, at least one e_i nonzero.
a(n), n>=1, is also generated by the following Dirichlet convolutions:
  a(n) = A157658(n) * A000012(n),
  a(n) = A008683(n) * A032741(n).
a(n) appears as a factor in the following Dirichlet convolutions:
  a(n) * A000010(n) = A051953(n),
  a(n) * A000027(n) = A001065(n),
  a(n) * A000012(n) = A032741(n).
a(n) is also both the number of disconnected 0-regular graphs on n vertices and the number of disconnected 1-regular graphs on 2n vertices. - Jason Kimberley, Sep 27 2011
Partial sums of A185012. - Jason Kimberley, Oct 15 2011
Decimal expansion of 1/900. - Elmo R. Oliveira, May 05 2024

J. S. Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000010, A000012, A000027, A000040, A001065, A008683, A032741, A051953, A057427, A157658, A185012.

PadRight[{0,0},120,{1}] (* Harvey P. Dale, Jun 03 2019 *)

a(n) = A057427(n-1) for n >= 2.
From Elmo R. Oliveira, Jul 20 2024: (Start)
G.f.: x^2/(1-x).
E.g.f.: exp(x) - x - 1. (End)

Definition simplified by R. J. Mathar, May 17 2010

A185983 Triangle read by rows: number of set partitions of n elements with k circular connectors.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 8, 4, 2, 1, 1, 20, 15, 14, 1, 1, 6, 53, 61, 68, 11, 3, 1, 25, 159, 267, 295, 97, 32, 1, 1, 93, 556, 1184, 1339, 694, 242, 28, 3, 1, 346, 2195, 5366, 6620, 4436, 1762, 371, 48, 2, 1, 1356, 9413, 25400, 34991, 27497, 12977, 3650, 634, 53, 3

Offset: 0

Brian Drake, Feb 08 2011

A pair (a,a+1) in a set partition with m blocks is a circular connector if a is in block i and a+1 is in block (i mod m)+1 for some i. In addition, (n,1) is considered a circular connector if n is in block m.

			For a(4,2) = 8, the set partitions are 1/234, 134/2, 124/3, 123/4, 12/34, 14/23, 1/24/3, and 13/2/4.
For a(5,1) = 1, the set partition is 13/25/4.
For a(6,6) = 3, the set partitions are 135/246, 14/25/36, 1/2/3/4/5/6.
Triangle begins:
  1;
  1, 0;
  1, 0,  1;
  1, 0,  3,  1;
  1, 0,  8,  4,  2;
  1, 1, 20, 15, 14,  1;
  1, 6, 53, 61, 68, 11, 3;
  ...

Alois P. Heinz, Rows n = 0..60, flattened
T. Mansour and A. O. Munagi, Block-connected set partitions, European J. Combin., 31 (2010), 887-902.

Cf. A185982.  Row sums are A000110.
T(n,n) = A032741(n) if n>0. - Alois P. Heinz, Oct 14 2011
T(2n,n) gives A362944.
Cf. A271272, A271273.

b:= proc(n, i, m, t) option remember; `if`(n=0, x^(t+
     `if`(i=m and m<>1, 1, 0)), add(expand(b(n-1, j,
      max(m, j), `if`(j=m+1, 0, t+`if`(j=1 and i=m
      and j<>m, 1, 0)))*`if`(j=i+1, x, 1)), j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1, 0$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 30 2016

b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, x^(t + If[i == m && m != 1, 1, 0]), Sum[Expand[b[n - 1, j, Max[m, j], If[j == m + 1, 0, t + If[j == 1 && i == m && j != m, 1, 0]]]*If[j == i + 1, x, 1]], {j, 1, m + 1}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 1, 0, 0] ];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 19 2016, after Alois P. Heinz *)

More terms from Alois P. Heinz, Oct 14 2011

A291127 Consider the zeros of the polynomial P(m,x) whose coefficients are the divisors of a number m. The sequence lists the numbers m such that P(m,x) contains at least two zeros that are purely imaginary numbers.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 42, 46, 51, 54, 55, 57, 58, 62, 65, 66, 69, 74, 77, 78, 82, 85, 86, 87, 88, 91, 93, 94, 95, 102, 104, 106, 110, 111, 114, 115, 118, 119, 122, 123, 125, 128, 129, 130, 133, 134, 136, 138, 141, 142, 143, 145

Offset: 1

Michel Lagneau, Aug 18 2017

nonn

P(m,x) = Sum_{i=1..k} d(i)*x^(i-1) where d(1), d(2), ..., d(k) are the k divisors of m.
The number of zeros of the polynomial P(n,x) is given by A032741(n).
We observe that all the zeros of the polynomial are located in the unit circle.
Conjecture: A032741(a(n))= p where p == 3 (mod 4), p prime. Examples:
A032741(m) = 3 for m = 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, ...
A032741(m) = 7 for m = 42, 54, 66, 78, 88, ...
A032741(m) = 11 for m = 156, 204, 228, 276, 294, 342, 348, 372, ...
A032741(m) = 19 for m = 2544, 2832, 2928, 3216, 3408, 3504, 3792, ...
A032741(m) = 23 for m = 24492, 25428, 26052, 26364, 26988, 27924, ...
Except for the two purely imaginary zeros of P(m,x), it seems that the complex zeros are of the form u +- u*i where u is a real number.
From Wolfdieter Lang, Nov 07 2017: (Start)
P(m,x) is the row m polynomial of A027750 with increasing powers of x.
The numbers m = 1 and m = prime obviously do not appear in this sequence. The composite numbers m belonging to irreducible polynomials P(m,x) over the integers given in A292226 also do not appear in this sequence. Moreover, the composite numbers m with factorizable P(m,x) without a factor of the type a*x^2 + b, with positive integers a and b, also do not appear in this sequence; these are the numbers 18, 20, 28, 32, 44, ... (End)
Are there numbers m with more than one pair of purely imaginary solutions? - Wolfdieter Lang, Nov 14 2017
From Robert Israel, Nov 14 2017: (Start)
The even and odd parts of P(m,x) are of the form A(x^2) and x*B(x^2) for polynomials A and B with integer coefficients, and pairs of imaginary roots of P(m,x) correspond to negative roots of the gcd of A and B.
Includes the following:
  p^k where p is prime and k==3 (mod 4).
  p*q^k where k is odd and p, q are prime with either p < q or p > q^k.
  p*q*r^k where p, q, r are distinct primes and r > p*q.
(End)

			42 is in the sequence because P(42,x) = 1 + 2x + 3x^2 + 6x^3 + 7x^4 + 14x^5 + 21x^6 + 42x^7 = (1 + 2*x)*(1 + 3*x^2)*(1 + 7*x^4), and the seven zeros are -1/2, +(1/3)*sqrt(3)*i, -(1/3)*sqrt(3)*i, r*(1+i), r*(1-i), r*(-1+i), r*(-1-i) with r = 7^(3/4)*sqrt(2)/14. The relevant factor for the two purely imaginary zeros is (1 + 3*x^2). - _Wolfdieter Lang_, Nov 13 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A027750, A032741, A292226.

F:= proc(n) local x,d,i,A,B,R;
  d:= sort(convert(numtheory:-divisors(n),list));
  A:= add(d[2*i]*x^(i-1),i=1..nops(d)/2);
  B:= add(d[2*i+1]*x^i,i=0..(nops(d)-1)/2);
  R:= gcd(A,B);
  sturm(sturmseq(R,x),x,-infinity,0) > 0;
end proc:
select(F, [$1..1000]); # Robert Israel, Nov 14 2017

Position[#, k_ /; k >= 2][[All, 1]] &@ Table[Count[Re /@ Values@ Apply[Join, Solve[Normal@ SeriesData[x, 0, #, 0, Length@ #, 1] == 0, x]], 0] &@ Divisors@ n, {n, 150}] (* Michael De Vlieger, Aug 21 2017 *)

isok(n) = {my(d = divisors(n), p = sum(k=1, #d, x^(k-1)*d[k])); #select(x->(real(x)==0), polroots(p)) >= 2;} \\ Michel Marcus, Sep 09 2017

cp's OEIS Frontend

A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

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A054009 n read modulo (number of proper divisors of n).

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A054015 a(n) is Chowla function of n read modulo (number of proper divisors of n), a(1) = 0 by convention.

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A063806 Numbers with a prime number of proper divisors.

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A069930 Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1.

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A084113 Number of multiplications when calculating A084110(n).

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A100577 Number of sets of divisors of n with an odd sum.

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A157928 a(n) = 0 if n < 2, = 1 otherwise.