A099777 -id:A099777 - cp's OEIS frontend

A372785 a(n) = tau(5n) = A000005(5n).

2, 4, 4, 6, 3, 8, 4, 8, 6, 6, 4, 12, 4, 8, 6, 10, 4, 12, 4, 9, 8, 8, 4, 16, 4, 8, 8, 12, 4, 12, 4, 12, 8, 8, 6, 18, 4, 8, 8, 12, 4, 16, 4, 12, 9, 8, 4, 20, 6, 8, 8, 12, 4, 16, 6, 16, 8, 8, 4, 18, 4, 8, 12, 14, 6, 16, 4, 12, 8, 12, 4, 24, 4, 8, 8, 12, 8, 16, 4, 15, 10, 8, 4, 24, 6, 8, 8, 16, 4, 18, 8, 12, 8, 8, 6

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A283118.

Cf. A000005, A099777, A372713, A372784, A372786, A372787, A372788, A372789, A372790, A372791, A372792.

Table[DivisorSigma[0, 5*n], {n, 1, 150}]

A372785(n) = numdiv(5*n); \\ Antti Karttunen, Jan 13 2025

Sum_{k=1..n} a(k) ~ (9*n*(log(n) + 2*gamma - 1) + n*log(5)) / 5, where gamma is the Euler-Mascheroni constant A001620.

More terms from Antti Karttunen, Jan 13 2025

A372787 a(n) = tau(7n) = A000005(7n).

Original entry on oeis.org

2, 4, 4, 6, 4, 8, 3, 8, 6, 8, 4, 12, 4, 6, 8, 10, 4, 12, 4, 12, 6, 8, 4, 16, 6, 8, 8, 9, 4, 16, 4, 12, 8, 8, 6, 18, 4, 8, 8, 16, 4, 12, 4, 12, 12, 8, 4, 20, 4, 12, 8, 12, 4, 16, 8, 12, 8, 8, 4, 24, 4, 8, 9, 14, 8, 16, 4, 12, 8, 12, 4, 24, 4, 8, 12, 12, 6, 16, 4

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A283078.

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372788, A372789, A372790, A372791, A372792.

Table[DivisorSigma[0, 7*n], {n, 1, 150}]

Sum_{k=1..n} a(k) ~ (13*n*(log(n) + 2*gamma - 1) + n*log(7)) / 7, where gamma is the Euler-Mascheroni constant A001620.

A372788 a(n) = tau(8n) = A000005(8n).

Original entry on oeis.org

4, 5, 8, 6, 8, 10, 8, 7, 12, 10, 8, 12, 8, 10, 16, 8, 8, 15, 8, 12, 16, 10, 8, 14, 12, 10, 16, 12, 8, 20, 8, 9, 16, 10, 16, 18, 8, 10, 16, 14, 8, 20, 8, 12, 24, 10, 8, 16, 12, 15, 16, 12, 8, 20, 16, 14, 16, 10, 8, 24, 8, 10, 24, 10, 16, 20, 8, 12, 16, 20, 8, 21

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A283122.

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372787, A372789, A372790, A372791, A372792.

Table[DivisorSigma[0, 8*n], {n, 1, 150}]

Sum_{k=1..n} a(k) ~ (20*n*(log(n) + 2*gamma - 1) + n*12*log(2)) / 8, where gamma is the Euler-Mascheroni constant A001620.

A372790 a(n) = tau(10n) = A000005(10n).

Original entry on oeis.org

4, 6, 8, 8, 6, 12, 8, 10, 12, 9, 8, 16, 8, 12, 12, 12, 8, 18, 8, 12, 16, 12, 8, 20, 8, 12, 16, 16, 8, 18, 8, 14, 16, 12, 12, 24, 8, 12, 16, 15, 8, 24, 8, 16, 18, 12, 8, 24, 12, 12, 16, 16, 8, 24, 12, 20, 16, 12, 8, 24, 8, 12, 24, 16, 12, 24, 8, 16, 16, 18, 8, 30

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372787, A372788, A372789, A372791, A372792.

Table[DivisorSigma[0, 10*n], {n, 1, 150}]

Sum_{k=1..n} a(k) ~ (27*n*(log(n) + 2*gamma - 1) + n*(9*log(2) + 3*log(5))) / 10, where gamma is the Euler-Mascheroni constant A001620.

A372791 a(n) = tau(11n) = A000005(11n).

Original entry on oeis.org

2, 4, 4, 6, 4, 8, 4, 8, 6, 8, 3, 12, 4, 8, 8, 10, 4, 12, 4, 12, 8, 6, 4, 16, 6, 8, 8, 12, 4, 16, 4, 12, 6, 8, 8, 18, 4, 8, 8, 16, 4, 16, 4, 9, 12, 8, 4, 20, 6, 12, 8, 12, 4, 16, 6, 16, 8, 8, 4, 24, 4, 8, 12, 14, 8, 12, 4, 12, 8, 16, 4, 24, 4, 8, 12, 12, 6, 16

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A099777, A372713, A372784, A372785, A372786, A372787, A372788, A372789, A372790, A372792.

Table[DivisorSigma[0, 11*n], {n, 1, 150}]

Sum_{k=1..n} a(k) ~ (21*n*(log(n) + 2*gamma - 1) + n*log(11)) / 11, where gamma is the Euler-Mascheroni constant A001620.

A129628 Inverse Moebius transform of A001511.

Original entry on oeis.org

1, 3, 2, 6, 2, 6, 2, 10, 3, 6, 2, 12, 2, 6, 4, 15, 2, 9, 2, 12, 4, 6, 2, 20, 3, 6, 4, 12, 2, 12, 2, 21, 4, 6, 4, 18, 2, 6, 4, 20, 2, 12, 2, 12, 6, 6, 2, 30, 3, 9, 4, 12, 2, 12, 4, 20, 4, 6, 2, 24, 2, 6, 6, 28, 4, 12, 2, 12, 4, 12, 2, 30, 2, 6, 6, 12, 4, 12, 2, 30, 5, 6, 2, 24, 4, 6, 4, 20

Offset: 1

Ralf Stephan, May 31 2007

Dirichlet convolution of A000005 with A209229. - Ridouane Oudra, Jul 25 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Row sums of A129353.
Cf. A000005, A001511, A001620, A129628.
Cf. A007814, A099777, A115364, A001227, A209229.

seq(add(padic[ordp](2*d, 2), d in numtheory[divisors](n)), n=1..100); # Ridouane Oudra, Sep 30 2024

f[p_, e_] := If[p==2, (e+1)*(e+2)/2, e+1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 30 2020 *)

a(n)={sumdiv(n, d, 1 + valuation(d, 2))} \\ Andrew Howroyd, Aug 04 2018

a(2n) = a(n) + A000005(2n), a(2n+1) = A000005(2n+1).
Dirichlet g.f.: zeta(s)^2 * 2^s/(2^s-1). - Ralf Stephan, Jun 17 2007, corrected by Vaclav Kotesovec, Feb 02 2019
a(n) = Sum_{d|n} A001511(d). - Andrew Howroyd, Aug 04 2018
Sum_{k=1..n} a(k) ~ 2*n * (2*gamma - 1 + log(n/2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = e+1 for p > 2. - Amiram Eldar, Sep 30 2020
From Ridouane Oudra, Sep 30 2024: (Start)
a(n) = Sum_{i=0..A007814(n)} tau(n/2^i).
a(n) = Sum_{d|2*n} A007814(d).
a(n) = (1/2)*A001511(n)*A099777(n).
a(n) = (1/2)*(A001511(n) + 1)*A000005(n).
a(n) = A115364(n)*A001227(n). (End)

A263084 a(n) = A263086(n) - A263085(n).

Original entry on oeis.org

1, 2, 4, 6, 7, 11, 13, 14, 18, 22, 22, 28, 29, 31, 37, 41, 41, 46, 48, 52, 58, 62, 60, 68, 71, 73, 79, 83, 83, 93, 95, 96, 100, 104, 108, 118, 120, 120, 124, 132, 131, 141, 141, 145, 155, 157, 157, 165, 169, 172, 178, 184, 180, 190, 196, 202, 208, 210, 208, 220, 221, 223, 231, 237, 241, 251, 251, 251, 257, 267, 267, 278

Offset: 1

Antti Karttunen, Oct 12 2015

nonn

A354452 Number of middle divisors of 2*n.

Original entry on oeis.org

1, 1, 2, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 1, 0, 2, 2, 0, 0, 2, 1, 0, 2, 2, 0, 2, 0, 1, 2, 0, 2, 3, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 1, 1, 0, 2, 0, 2, 2, 2, 0, 0, 0, 4, 0, 0, 2, 1, 2, 2, 0, 0, 0, 2, 0, 3, 0, 0, 2, 0, 2, 2, 0, 2, 1, 0, 0, 2, 2, 0, 0, 2, 0, 4, 2, 0, 0, 0, 2, 2, 0, 1, 2, 1, 0, 2, 0, 2, 2

Offset: 1

Omar E. Pol, May 30 2022

nonn

a(n) is the number of middle divisors of the n-th even number.
a(n) is also the width of the terrace at the level 2*n 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). For more information about the subparts see A279387.

Antti Karttunen, Table of n, a(n) for n = 1..32768

Bisection of A067742.

Cf. A005843, A099777, A237048, A237270, A237271, A237593, A245092, A249351, A279387, A319796, A354451.

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

A354452(n) = { n <<= 1; sumdiv(n, d, my(d2 = d^2); (n / 2 < d2 && d2 <= n << 1)); }; \\ Antti Karttunen, Jan 17 2025, after program in A067742 by M. F. Hasler, May 12 2008

a(n) = A067742(2n).

a(n) = A067742(A005843(n)).

A363132 Number of integer partitions of 2n such that 2*(minimum) = (mean).

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 15, 14, 32, 34, 65, 55, 150, 100, 225, 237, 425, 296, 824, 489, 1267, 1133, 1809, 1254, 4018, 2142, 4499, 4550, 7939, 4564, 14571, 6841, 18285, 16047, 23408, 17495, 52545, 21636, 49943, 51182, 92516, 44582, 144872, 63260, 175318, 169232, 205353

Offset: 0

Gus Wiseman, May 23 2023

nonn

Equivalently, n = (length)*(minimum).

			The a(2) = 1 through a(7) = 14 partitions:
  (31)  (321)  (62)    (32221)  (93)      (3222221)
        (411)  (3221)  (33211)  (552)     (3322211)
               (3311)  (42211)  (642)     (3332111)
               (4211)  (43111)  (732)     (4222211)
               (5111)  (52111)  (822)     (4322111)
                       (61111)  (322221)  (4331111)
                                (332211)  (4421111)
                                (333111)  (5222111)
                                (422211)  (5321111)
                                (432111)  (5411111)
                                (441111)  (6221111)
                                (522111)  (6311111)
                                (531111)  (7211111)
                                (621111)  (8111111)
                                (711111)

Removing the factor 2 gives A099777.
Taking maximum instead of mean and including odd indices gives A118096.
For length instead of mean and including odd indices we have A237757.
For (maximum) = 2*(mean) see A361851, A361852, A361853, A361854, A361855.
For median instead of mean we have A361861.
These partitions have ranks A363133.
For maximum instead of minimum we have A363218.
For median instead of minimum we have A363224.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
A268192 counts partitions by complement size, ranks A326844.
Cf. A053263, A111907, A237753, A237755, A237824, A327482, A349156, A361906, A361907, A363134.

Table[Length[Select[IntegerPartitions[2n],2*Min@@#==Mean[#]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A363132(n): return sum(1 for s,p in partitions(n<<1,m=n,size=True) if n==s*min(p,default=0)) if n else 0 # Chai Wah Wu, Sep 21 2023

a(31)-a(46) from Chai Wah Wu, Sep 21 2023

A372674 a(n) = Sum_{j=1..n} Sum_{k=1..n} tau(j*k).

Original entry on oeis.org

1, 8, 23, 54, 89, 162, 221, 326, 439, 596, 707, 964, 1107, 1352, 1645, 1976, 2179, 2630, 2865, 3390, 3859, 4316, 4615, 5406, 5883, 6444, 7059, 7892, 8299, 9430, 9877, 10794, 11635, 12424, 13361, 14852, 15415, 16324, 17349, 18952, 19587, 21342, 22017, 23486, 25177

Offset: 1

Vaclav Kotesovec, May 10 2024

nonn

For m>=1, Sum_{j=1..n} tau(m*j) = A018804(m) * n * log(n) + O(n).

If p is prime, then Sum_{j=1..n} tau(p*j) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n) / (n^2 * (log(n) + 2*gamma - 1/2)^2) for n = 1..100000

Cf. A372633, A372675.
Cf. A006218, A263086.
cf. A000005, A099777, A372713.

Table[Sum[DivisorSigma[0, j*k], {j, 1, n}, {k, 1, n}], {n, 1, 50}]
s = 1; Join[{1}, Table[s += DivisorSigma[0, n^2] + 2*Sum[DivisorSigma[0, j*n], {j, 1, n - 1}], {n, 2, 50}]]

cp's OEIS Frontend

A372785 a(n) = tau(5*n) = A000005(5*n).

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A372787 a(n) = tau(7*n) = A000005(7*n).

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A372788 a(n) = tau(8*n) = A000005(8*n).

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A372790 a(n) = tau(10*n) = A000005(10*n).

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A372791 a(n) = tau(11*n) = A000005(11*n).

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A129628 Inverse Moebius transform of A001511.

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A263084 a(n) = A263086(n) - A263085(n).

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A354452 Number of middle divisors of 2*n.

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A372785 a(n) = tau(5n) = A000005(5n).

A372787 a(n) = tau(7n) = A000005(7n).

A372788 a(n) = tau(8n) = A000005(8n).

A372790 a(n) = tau(10n) = A000005(10n).

A372791 a(n) = tau(11n) = A000005(11n).