A099774 -id:A099774 - cp's OEIS frontend

A274658 Irregular triangle which lists in row n the divisors of 2*n+1.

1, 1, 3, 1, 5, 1, 7, 1, 3, 9, 1, 11, 1, 13, 1, 3, 5, 15, 1, 17, 1, 19, 1, 3, 7, 21, 1, 23, 1, 5, 25, 1, 3, 9, 27, 1, 29, 1, 31, 1, 3, 11, 33, 1, 5, 7, 35, 1, 37, 1, 3, 13, 39, 1, 41, 1, 43, 1, 3, 5, 9, 15, 45

Offset: 0

Wolfdieter Lang, Jul 18 2016

The length of row n is A099774(n+1).
This gives the odd numbered rows of the irregular triangle A027750.
The row sums are given in A008438.
The entries of row n appear, for instance, as arguments of sin in the Fourier expansion of Jacobi's elliptic function sn in the second factor Sum_{n>=0} (q^n/(1-q^(2*n+1)))*sin((2*n+1)*v) as coefficients of q^n. See e.g., the formula in Abramowitz-Stegun, p. 575, 16.23.1 (or 16.23.2 for cn but with signs). See also A274659.

			The irregular triangle T(n, k) begins:
  n, 2n+1\k 1  2   3   4 ...
  0,   1:   1
  1,   3:   1  3
  2,   5:   1  5
  3,   7:   1  7
  4,   9:   1  3   9
  5,  11:   1 11
  6,  13:   1 13
  7,  15:   1  3   5  15
  8,  17:   1 17
  9,  19:   1 19
  10, 21:   1  3   7  21
  11, 23:   1 23
  12, 25:   1  5  25
  13, 27:   1  3   9  27
  14, 29:   1 29
  15, 31:   1 31
  16, 33:   1  3  11  33
  17, 35:   1  5   7  35
  18, 37:   1 37
  19, 39:   1  3  13  39
  20, 41:   1 41
...
The above mentioned second factor in the sn formula has as q^4 coefficient: sin(1*v) + sin(3*v) + sin(9*v).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972,

Cf. A008438 (row sums), A027750, A099774 (row lengths), A274659.

Table[Divisors[2 n + 1], {n, 0, 22}] // Flatten (* Michael De Vlieger, Jul 18 2016 *)

row(n) = divisors(2*n+1); \\ Amiram Eldar, May 02 2025

T(n, k) = k-th divisor of 2*n+1 in increasing order.

A274660 Irregular triangle read by rows in which row n lists the divisors d of 2*n+1 (A274658), given the sign (-1)^(n + (d-1)/2).

Original entry on oeis.org

1, -1, 3, 1, 5, -1, 7, 1, -3, 9, -1, 11, 1, 13, -1, 3, -5, 15, 1, 17, -1, 19, 1, -3, -7, 21, -1, 23, 1, 5, 25, -1, 3, -9, 27, 1, 29, -1, 31, 1, -3, -11, 33, -1, -5, 7, 35, 1, 37, -1, 3, -13, 39, 1, 41, -1, 43, 1, -3, 5, 9, -15, 45, -1, 47, 1, -7, 49, -1, 3, -17, 51, 1, 53, -1, -5, 11, 55, 1, -3, -19, 57, -1, 59, 1, 61

Offset: 0

Wolfdieter Lang, Jul 27 2016

The length of row n is A099774(n+1).
The unsigned irregular triangle is given in A274658.
The sum of row n gives A228443(n).
The entries of row n appear in the Fourier expansion of Jacobi's elliptic function cn in the rewritten second factor Sum_{n>=0} (q^n/(1+q^(2*n+1))) * cos((2*n+1)*v) as Sum_{n>=0} q^n*Sum_{k=1..A099774(n+1)} sign(a(n,k))*cos(abs(a(n,k))*v). See e.g., the formula in Abramowitz-Stegun, p. 575, 16.23.2.

			The irregular triangle T(n, k) begins:
n, 2n+1\k  1  2   3   4 ...
0,   1:    1
1,   3:   -1  3
2,   5:    1  5
3,   7:   -1  7
4,   9:    1 -3   9
5,  11:   -1 11
6,  13:    1 13
7,  15:   -1  3  -5  15
8,  17:    1 17
9,  19:   -1 19
10, 21:    1 -3  -7  21
11, 23:   -1 23
12, 25:    1  5  25
13, 27:   -1  3  -9  27
14, 29:    1 29
15, 31:   -1 31
16, 33:    1 -3 -11  33
17, 35:   -1 -5   7  35
18, 37:    1 37
19, 39:   -1  3 -13  39
20, 41:    1 41
...
The above mentioned expansion coefficient of q^4 of the second factor of the cn formula is +cos(1*v) - cos(3*v) + cos(9*v).

Michael De Vlieger, Table of n, a(n) for n = 0..12574 (0 <= n <= 2500)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972,

Cf. A099774, A228443, A274658.

Table[(-1)^(n + (# - 1)/2) # &@ Divisors[2 n + 1], {n, 0, 30}] // Flatten (* Michael De Vlieger, Aug 01 2016 *)

T(n, k) = (-1)^(n + (d(k)-1)/2)*d(k) with d(k) the k-th divisor of 2*n+1 in increasing order.

A274661 Triangle read by rows: T(n, m) gives the m-th contribution T(n, m)cos((2m+1)v) to the coefficient of q^n in the Fourier expansion of Jacobi's elliptic cn(u|k) function when expressed in the variables v = u/(2K(k)/Pi) and q, the Jacobi nome, written as series in (k/4)^2. K is the real quarter period of elliptic functions.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 1, -2, 0, 1, 2, -1, -2, 0, 1, -2, 3, 0, -2, 0, 1, -4, 2, 3, 0, -2, 0, 1, 4, -5, -1, 3, 0, -2, 0, 1, 7, -3, -6, 0, 3, 0, -2, 0, 1, -7, 9, 2, -6, 0, 3, 0, -2, 0, 1, -11, 5, 11, -1, -6, 0, 3, 0, -2, 0, 1, 11, -15, -3, 11, 0, -6, 0, 3, 0, -2, 0, 1, 17, -9, -17, 2, 11, 0, -6, 0, 3, 0, -2, 0, 1, -17, 23, 6, -18, -1, 11, 0, -6, 0, 3, 0, -2, 0, 1

Offset: 0

Wolfdieter Lang, Jul 27 2016

If one takes the row polynomials as P(n, x) = Sum_{m=0..n} T(n, m)*x^m, n >= 0, Jacobi's elliptic function cn(u|k) in terms of the new variables v and q becomes cn(u|k) = Sum_{n>=0} P(n, x)*q^n, if in P(n, x) one replaces x^j by cos((2*j+1)*v).
v=v(u,k^2) and q=q(k^2) are computed with the help of A038534/A056982 for (2/Pi)*K and A002103 for q expanded in powers of (k/4)^2.
A test for cn(u|k) with u = 1, k = sqrt(1/2), that is v approximately 0.8472130848 and q approximately 0.04321389673, with rows n=0..10 (q powers not exceeding 10) gives 0.5959766014 to be compared with cn(1|sqrt(1/2)) approximately 0.5959765676.
For the derivation of the Fourier series formula of cn given in Abramowitz-Stegun (but there the notation sn(u|m=k^2) is used for sn(u|k)) see, e.g., Whittaker and Watson, p. 511 or Armitage and Eberlein, Exercises on p. 55.
For sn see A274659 (differently signed triangle).
The sum of entries in row n is P(n, 1) = A000007(n): 1, repeat 0. Proof: due to the g.f. identity (from the convolution)
  Sum_{n >= 0} x^n/(1 + x^(2*n+1))  = (Sum_{n >= 0} x^(n*(n+1)))^2.
  This is proved by bisecting the g.f. on the l.h.s. which generates c(n, 1) = (-1)^n*Sum_{2*r+1 | 2*n+1} (-1)^n. The part with n = 2*k+1 vanishes due to r_2(4*k+1)/4 = 0, where r_2(n) is the number of solutions of n as a sum of two squares. See the Grosswald reference. The part with n = 2*k becomes Sum_{k >= 0} x^(2*k) r_2(4*k+1)/4 which is the r.h.s. See A008441, the Broadhurst Oct 20 2002 comment.
For another version of this expansion of cn see A275791.
See also the W. Lang link, eqs. (43) and (44). - Wolfdieter Lang, Aug 26 2016

			The triangle T(n, m) begins:
      m  0   1  2  3  4  5  6  7  8  9 10 11
n\ 2m+1  1   3  5  7  9 11 13 15 17 19 21 23
0:       1
1:      -1   1
2:      -1   0  1
3:       1  -2  0  1
4:       2  -1 -2  0  1
5:      -2   3  0 -2  0  1
6:      -4   2  3  0 -2  0  1
7:       4  -5 -1  3  0 -2  0  1
8:       7  -3 -6  0  3  0 -2  0  1
9:      -7   9  2 -6  0  3  0 -2  0  1
10:    -11   5 11 -1 -6  0  3  0 -2  0  1
11:     11 -15 -3 11  0 -6  0  3  0 -2  0  1
...
n = 4: c(0, x)*a(4) + c(2, x)*a(2) + c(4, x)*a(0) = (+x^1)*3 +  (+x^1 + x^5)*(-2) + (+x^1 - x^3 + x^9)*1 = +2*x^1 - x^3 - 2*x^5 + 0*x^7 + x^9. Hence row n=4 is 2, -1, -2, 0, 1.
From A274660, row n = 4: c(4, x) = +x^1 - x^3 +x^9.
n = 4: P(4, x) = 2 - 1*x^1 - 2*x^2 + 1*x^4, that is the contribution of order q^4 to cn in the new variables is (2*cos(v)  - 1*cos(3*v) - 2*cos(5*v) + 1*cos(9*v))*q^4.

J. V. Armitage and W. F. Eberlein, Elliptic Functions, London Mathematical Society, Student Texts 67, Cambridge University Press, 2006.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15, Theorem 3.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 375, 16.23.2.
Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A000007, A099774, A274621, A274659, A274660, A275791.

T(n, m) = [x^(2*m+1)]Sum_{j=0..n} c(j, x)*a(n-j), with a(k) = A274621(k/2) if k is even and a(k) = 0 if k is odd, and c(j, x) = (-1)^j*Sum_{2*r+1 | 2*j+1} (-1)^r*x^(2*r+1) = Sum_{k=1..A099774(j+1)} sign(A274660(j, k))*x^(abs(A274660(j, k))), for j >= 0.

A366441 The number of divisors of the 5-rough numbers (A007310).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 3, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 2, 4, 2, 2, 3, 2, 6, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 2, 4, 4, 4, 2, 2, 2, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4

Offset: 1

Amiram Eldar, Oct 10 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001620, A007310, A366442.

Similar sequences: A048691, A072048, A076400, A099774, A358040, A363194, A363195.

a[n_] := DivisorSigma[0, 2*Floor[3*n/2] - 1]; Array[a, 100]

PARI
```
a(n) = numdiv((3*n)\2 << 1 - 1)
```

from sympy import divisor_count
def A366441(n): return divisor_count((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

a(n) = A000005(A007310(n)).

Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - 1 + 2*log(6)) * n / 3, where gamma is Euler's constant (A001620).

A372714 a(n) = tau(3n-1) = A000005(3n-1).

Original entry on oeis.org

2, 2, 4, 2, 4, 2, 6, 2, 4, 2, 6, 4, 4, 2, 6, 2, 6, 2, 8, 2, 4, 4, 6, 2, 4, 4, 10, 2, 4, 2, 6, 4, 6, 2, 8, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 2, 12, 4, 4, 2, 8, 4, 4, 4, 6, 2, 8, 2, 10, 2, 8, 4, 6, 2, 4, 2, 12, 4, 4, 4, 6, 4, 4, 4, 12, 2, 8, 2, 6, 2, 6, 6, 8, 2, 4, 2

Offset: 1

Vaclav Kotesovec, May 11 2024

nonn

Cf. A372713, A372715.

Cf. A000005, A099777, A099774.

Table[DivisorSigma[0, 3*n-1], {n, 1, 150}]

Sum_{k=1..n} a(k) ~ 2*n * (log(n) + 2*gamma - 1 + 2*log(3)) / 3, where gamma is the Euler-Mascheroni constant A001620.

A372715 a(n) = tau(3n-2) = A000005(3n-2).

Original entry on oeis.org

1, 3, 2, 4, 2, 5, 2, 4, 3, 6, 2, 4, 2, 8, 2, 4, 3, 6, 4, 4, 2, 7, 2, 8, 2, 6, 2, 4, 4, 8, 4, 4, 2, 9, 2, 4, 2, 10, 4, 4, 3, 6, 2, 8, 4, 8, 2, 4, 4, 6, 2, 8, 2, 12, 2, 4, 3, 6, 6, 4, 2, 8, 4, 8, 2, 9, 2, 4, 4, 10, 2, 4, 4, 12, 2, 4, 2, 8, 4, 8, 2, 6, 4, 8, 4, 9

Offset: 1

Vaclav Kotesovec, May 11 2024

nonn

Cf. A372713, A372714.

Cf. A000005, A099777, A099774.

Table[DivisorSigma[0, 3*n-2], {n, 1, 150}]

Sum_{k=1..n} a(k) ~ 2*n * (log(n) + 2*gamma - 1 + 2*log(3)) / 3, where gamma is the Euler-Mascheroni constant A001620.

A378436 Inverse Möbius transform of A033630, where A033630 is the number of partitions of n into distinct divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 4, 2, 9, 2, 4, 4, 5, 2, 9, 2, 7, 4, 4, 2, 16, 3, 4, 4, 7, 2, 12, 2, 6, 4, 4, 4, 21, 2, 4, 4, 12, 2, 11, 2, 6, 6, 4, 2, 28, 3, 6, 4, 6, 2, 14, 4, 11, 4, 4, 2, 53, 2, 4, 6, 7, 4, 11, 2, 6, 4, 8, 2, 60, 2, 4, 6, 6, 4, 10, 2, 20, 5, 4, 2, 43, 4, 4, 4, 9, 2, 41, 4, 6, 4, 4, 4, 51, 2, 6, 6, 12

Offset: 1

Antti Karttunen, Nov 26 2024

nonn

The odd bisection differs from A099774 (the odd bisection of A000005) apparently in the positions given by A005231 (odd abundant numbers): 945, 1575, 2205, 2835, 3465, ...

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

Cf. A000005, A005231, A033630, A099774, A378438 (Dirichlet inverse).

A033630(n) = if(!n, 1, my(p=1); fordiv(n, d, p *= (1 + 'x^d)); polcoeff(p, n));
A378436(n) = sumdiv(n,d,A033630(d));

a(n) = Sum_{d|n} A033630(d).

A354451 Number of middle divisors of 2*n-1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Omar E. Pol, May 30 2022

nonn

a(n) is the number of middle divisors of the n-th odd number.
a(n) is also the width of the terrace at the level 2*n-1 starting from the top in the main diagonal of the stepped pyramid described in A245092.
a(n) is also the number of central subparts in the symmetric representation of sigma(2n-1). For more information about the subparts see A279387.

Bisection of A067742.

Cf. A005408, A099774, A237048, A237270, A237271, A237593, A245092, A249351, A279387, A319529, A354452.

a[n_] := DivisorSum[2*n - 1, 1 &, n <= #^2 < 4*n - 2 &]; Array[a, 100] (* Amiram Eldar, Jun 01 2022 *)

a(n) = A067742(2n-1).

a(n) = A067742(A005408(n-1)).

A334764 a(n) = Product_{k=1..n} d(2*k - 1), where d() is the number of divisors function A000005.

Original entry on oeis.org

1, 2, 4, 8, 24, 48, 96, 384, 768, 1536, 6144, 12288, 36864, 147456, 294912, 589824, 2359296, 9437184, 18874368, 75497472, 150994944, 301989888, 1811939328, 3623878656, 10871635968, 43486543872, 86973087744, 347892350976, 1391569403904, 2783138807808, 5566277615616, 33397665693696, 133590662774784

Offset: 1

Ctibor O. Zizka, May 10 2020

nonn

			a(3) = d(1)*d(3)*d(5) = 1*2*2 = 4.

Cf. A000005, A066843, A099774.

Rest @ FoldList[Times, 1, DivisorSigma[0, Range[1, 50, 2]]] (* Amiram Eldar, May 10 2020 *)

a(n) = prod(k=1, n, numdiv(2*k-1)); \\ Michel Marcus, May 10 2020

cp's OEIS Frontend

A274658 Irregular triangle which lists in row n the divisors of 2*n+1.

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A274660 Irregular triangle read by rows in which row n lists the divisors d of 2*n+1 (A274658), given the sign (-1)^(n + (d-1)/2).

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A366441 The number of divisors of the 5-rough numbers (A007310).

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A372714 a(n) = tau(3*n-1) = A000005(3*n-1).

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A372715 a(n) = tau(3*n-2) = A000005(3*n-2).

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A378436 Inverse Möbius transform of A033630, where A033630 is the number of partitions of n into distinct divisors of n.

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A354451 Number of middle divisors of 2*n-1.

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A334764 a(n) = Product_{k=1..n} d(2*k - 1), where d() is the number of divisors function A000005.

A372714 a(n) = tau(3n-1) = A000005(3n-1).

A372715 a(n) = tau(3n-2) = A000005(3n-2).