A064840 -id:A064840 - cp's OEIS frontend

A064948 a(n) = Sum_{i|n, j|n} max(i,j).

1, 7, 10, 27, 16, 64, 22, 83, 55, 102, 34, 236, 40, 140, 140, 227, 52, 343, 58, 372, 192, 216, 70, 708, 141, 254, 244, 510, 88, 866, 94, 579, 296, 330, 296, 1241, 112, 368, 348, 1104, 124, 1184, 130, 786, 728, 444, 142, 1908, 267, 877, 452, 924, 160, 1504, 456

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

			a(6) = dot_product(1,3,5,7)*(1,2,3,6) = 1*1 + 3*2 + 5*3 + 7*6 = 64.

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

Cf. A000005, A000203, A064944, A064840.

with(numtheory): seq(add((2*i-1)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

A064948[n_] := #.(2*Range[Length[#]] - 1) & [Divisors[n]];
Array[A064948, 100] (* Paolo Xausa, Aug 14 2025 *)

a(n) = { my(d=divisors(n)); sum(i=1, #d, (2*i - 1)*d[i]) } \\ Harry J. Smith, Oct 01 2009

a(n) = Sum_{i=1..tau(n)} (2*i-1)*d_i, where {d_i}, i=1..tau(n), is the increasing sequence of the divisors of n.
a(n) = 2*A064944(n) - A000203(n). - Amiram Eldar, Dec 23 2024
From Ridouane Oudra, Aug 07 2025: (Start)
a(n) = A064944(n) + A064946(n).
a(n) = 2*A064946(n) + A000203(n).
a(n) = 2*A064840(n) - A064949(n). (End)

A071707 Numbers k that divide tau(k)*sigma(k).

Original entry on oeis.org

1, 2, 6, 12, 18, 24, 28, 40, 84, 120, 224, 234, 240, 252, 360, 468, 496, 672, 864, 936, 1638, 1920, 2016, 2480, 3276, 4320, 4680, 6048, 6528, 6552, 7440, 8128, 9360, 10880, 22320, 22932, 26208, 30240, 32640, 32760, 47616, 56896, 58752, 65520, 74880, 79360, 84480

Offset: 1

Reinhard Zumkeller, Jun 03 2002

nonn

			The divisors of 18 are {1,2,3,6,9,18}, so tau(18) = 6 and sigma(18) = 1+2+3+6+9+18 = 39, 18 is a term as 18*13 = 6*39 = tau(18)*sigma(18).

Donovan Johnson, Table of n, a(n) for n = 1..300

A007691 is a subsequence.

Cf. A064840 (A000005(n)*A000203(n)).

Select[Range[10^5], Divisible[Times @@ DivisorSigma[{0, 1}, #], #] &] (* Amiram Eldar, Apr 16 2025 *)

isok(k) = {my(f = factor(k)); !((numdiv(f) * sigma(f)) % k);} \\ Amiram Eldar, Apr 16 2025

A233541 a(n) = sigma(n) + phi(n) + d(n).

Original entry on oeis.org

3, 6, 8, 12, 12, 18, 16, 23, 22, 26, 24, 38, 28, 34, 36, 44, 36, 51, 40, 56, 48, 50, 48, 76, 54, 58, 62, 74, 60, 88, 64, 85, 72, 74, 76, 112, 76, 82, 84, 114, 84, 116, 88, 110, 108, 98, 96, 150, 102, 119, 108, 128, 108, 146, 116, 152, 120, 122, 120, 196, 124

Offset: 1

Wesley Ivan Hurt, Dec 12 2013

a(n) is the sum of the divisors of n plus the number of positive integers less than or equal to n and relatively prime to n plus the number of divisors of n.

If n is a prime, then a(n) = A064840(n). If n is a prime or a semiprime, then a(n) = 2(d(n) + n - 1).

			a(6) = 18; sigma(6) + phi(6) + d(6) = 12 + 2 + 4 = 18.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A064840.

with(numtheory); A233541:=n->sigma(n) + phi(n) + tau(n); seq(A233541(n), n=1..100);

Table[DivisorSigma[0,n] + DivisorSigma[1,n] + EulerPhi[n], {n,100}]

a(n) = sigma(n) + eulerphi(n) + numdiv(n); \\ Michel Marcus, Dec 07 2016

a(n) = A000203(n) + A000010(n) + A000005(n).

Dirichlet g.f.: (zeta(s)^3 + zeta(s-1)*zeta(s)^2 + zeta(s-1))/zeta(s). - Ilya Gutkovskiy, Dec 07 2016

A328485 Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-1).

Original entry on oeis.org

1, 4, 5, 9, 7, 20, 9, 18, 15, 28, 13, 45, 15, 36, 35, 35, 19, 60, 21, 63, 45, 52, 25, 90, 33, 60, 43, 81, 31, 140, 33, 68, 65, 76, 63, 135, 39, 84, 75, 126, 43, 180, 45, 117, 105, 100, 49, 175, 59, 132, 95, 135, 55, 172, 91, 162, 105, 124, 61, 315, 63, 132, 135, 133, 105

Offset: 1

Ilya Gutkovskiy, Oct 16 2019

Inverse Moebius transform of A034448.

Dirichlet convolution of A055615 with A064840.

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

Cf. A000005, A000203, A007429, A008683, A034448, A048691, A055615, A064840, A319131, A319132.

with(numtheory):
a:= n-> add(mobius(d)*tau(n/d)*sigma(n/d)*d, d=divisors(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 16 2019

Table[n DivisorSum[n, MoebiusMu[n/#] DivisorSigma[0, #] DivisorSigma[1, #]/# &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[DivisorSum[k, # &, CoprimeQ[#, k/#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (p^(e + 1) - p)/(p - 1) + e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2023 *)

a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, (p[i]^(e[i] + 1) - p[i])/(p[i] - 1) + e[i] + 1);} \\ Amiram Eldar, Feb 10 2023

G.f.: Sum_{k>=1} usigma(k) * x^k / (1 - x^k), where usigma = A034448.
a(n) = Sum_{d|n} usigma(d).
a(n) = n * Sum_{d|n} mu(n/d) * tau(d) * sigma(d) / d, where mu = A008683, tau = A000005 and sigma = A000203.
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (72 * zeta(3)). - Vaclav Kotesovec, Oct 17 2019
From Amiram Eldar, Feb 10 2023: (Start)
a(n) = Sum_{d|n} Sum_{d'|n, gcd(d, d')=1} d'.
Multiplicative with a(p^e) = (p^(e+1)-p)/(p-1) + e + 1. (End)

A087801 Greatest common divisor of tau(n)+sigma(n) and tau(n)*sigma(n), where tau = A000005 and sigma = A000203.

Original entry on oeis.org

1, 1, 2, 1, 4, 16, 2, 1, 1, 2, 2, 2, 4, 4, 4, 1, 4, 9, 2, 12, 4, 8, 2, 4, 1, 2, 4, 2, 4, 16, 2, 3, 4, 2, 4, 1, 4, 16, 4, 2, 4, 8, 2, 18, 12, 4, 2, 2, 3, 9, 4, 4, 4, 64, 4, 64, 4, 2, 2, 36, 4, 4, 2, 1, 8, 8, 2, 12, 4, 8, 2, 9, 4, 2, 2, 2, 4, 16, 2, 4, 1, 2, 2, 4, 16, 8, 4, 4, 4, 6, 4, 6, 4, 4, 4, 24

Offset: 1

Reinhard Zumkeller, Oct 11 2003

nonn

a(n) = GCD(A007503(n), A064840(n)).

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

Cf. A009191, A009194, A009205, A007503, A064840, A286360.

GCD[Total[#],Times@@#]&/@Table[{DivisorSigma[0,n],DivisorSigma[1,n]},{n,100}] (* Harvey P. Dale, Jul 17 2018 *)

A087801(n) = gcd(sigma(n)+numdiv(n), sigma(n)*numdiv(n)); \\ Antti Karttunen, May 22 2017

A126775 a(n) = phi(n)^2 * d(n) = A000010(n)^2 * A000005(n).

Original entry on oeis.org

1, 2, 8, 12, 32, 16, 72, 64, 108, 64, 200, 96, 288, 144, 256, 320, 512, 216, 648, 384, 576, 400, 968, 512, 1200, 576, 1296, 864, 1568, 512, 1800, 1536, 1600, 1024, 2304, 1296, 2592, 1296, 2304, 2048, 3200, 1152, 3528, 2400, 3456, 1936, 4232, 2560, 5292, 2400

Offset: 1

Jonathan Vos Post, May 27 2007

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

Cf. A000005, A000010, A064840, A127473, A256392.

[ EulerPhi(n)*EulerPhi(n)*NumberOfDivisors(n) : n in [1..100] ];

Table[EulerPhi[n]^2 DivisorSigma[0,n],{n,50}] (* Harvey P. Dale, Dec 05 2012 *)

Multiplicative with a(p^e) = (e+1)*(p-1)^2*p^(2*e-2). - Amiram Eldar, Dec 29 2022
From Vaclav Kotesovec, May 31 2024: (Start)
Dirichlet g.f.: zeta(s-2)^2 * Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).
Sum_{k=1..n} a(k) ~ f(3) * n^3 * (log(n) + 2*gamma - 1/3 + f'(3)/f(3)) / 3, where
f(3) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264...,
f'(3) = f(3) * Sum_{p prime} 2*(2*p - 1) * log(p) / (p^3 + p^2 - 3*p + 1) = f(3) * 1.6860441157206199528397247528679297282000614932962665074593283751342385...
and gamma is the Euler-Mascheroni constant A001620. (End)

A327830 Numbers m such that the geometric mean of tau(m) and sigma(m) is an integer.

Original entry on oeis.org

1, 7, 17, 22, 30, 31, 71, 94, 97, 115, 119, 127, 138, 154, 164, 165, 199, 210, 214, 217, 232, 241, 260, 265, 318, 337, 343, 374, 382, 449, 497, 510, 513, 517, 527, 577, 647, 658, 668, 679, 682, 705, 745, 759, 805, 862, 881, 889, 894, 930, 966, 967, 996, 1102, 1125

Offset: 1

Bernard Schott, Sep 27 2019

nonn

The first 20 terms of this sequence are also the first 20 terms of A144695: m such that sigma(m)/tau(m) is a square. Indeed, if sigma(m)/tau(m) is a square then sigma(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A327831; the first one is a(21) = 232.
The primes p of the form 2*k^2 - 1: 7, 17, 31, 71, ... (A066436) form a subsequence because sigma(p) * tau(p) = (2*k)^2.
Another subsequence consists of the terms m such that sigma(m) and tau(m) are both squares; this occurs when m is the product of two distinct primes p*q, p < q where sigma(m) = (p+1)*(q+1) is a square and tau(m) = 4. The first few terms are 22, 94, 115, 119, 214, ... They are in A256152.

			sigma(30) = 72 and tau(30) = 8, sigma(30)*tau(30) = 576 = 24^2, hence 30 is a term.

Cf. A000005 (tau), A000203 (sigma), A064840 (tau*sigma).
Cf. A011257 (similar, with phi(m) and sigma(m)), A144695 (sigma(m)/tau(m) is a square), A327831 (sigma(m) * tau(m) is a square but sigma(m)/tau(m) is not an integer).
Subsequences: A066436, A256152.

[k:k in [1..1150]| IsSquare(#Divisors(k)*DivisorSigma(1,k))]; // Marius A. Burtea, Sep 27 2019

filter:= s -> issqr(sigma(s)*tau(s)) : select(filter, [$1..2500]);

Select[Range[1000], IntegerQ @ Sqrt[DivisorSigma[0, #] * DivisorSigma[1, #]] &] (* Amiram Eldar, Sep 27 2019 *)

isok(m) = issquare(numdiv(m)*sigma(m)); \\ Michel Marcus, Sep 27 2019

A216365 Numbers n such that tau(n)*sigma(n) sets a new record.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, 840, 1008, 1080, 1200, 1260, 1440, 1680, 2100, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040

Offset: 1

Arkadiusz Wesolowski, Sep 05 2012

nonn

Positions of record values in A064840.

Not identical to A067128; e.g. a(22) = 144 < 168 = A067128(22).

Charles R Greathouse IV and Donovan Johnson, Table of n, a(n) for n = 1..400 (first 268 terms from Charles R Greathouse IV)

Cf. A000005, A000203, A064840.

lst = {}; k = 0; Do[n = DivisorSigma[0, i]*DivisorSigma[1, i]; If[n > k, AppendTo[lst, i]; k = n], {i, 7!}]; lst

r=0;for(n=1,1e9,t=numdiv(n)*sigma(n);if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Sep 05 2012

A309153 a(n) = A000203(n)*A001227(n).

Original entry on oeis.org

1, 3, 8, 7, 12, 24, 16, 15, 39, 36, 24, 56, 28, 48, 96, 31, 36, 117, 40, 84, 128, 72, 48, 120, 93, 84, 160, 112, 60, 288, 64, 63, 192, 108, 192, 273, 76, 120, 224, 180, 84, 384, 88, 168, 468, 144, 96, 248, 171, 279, 288, 196, 108, 480, 288, 240, 320, 180, 120, 672, 124, 192, 624, 127, 336, 576, 136, 252, 384

Offset: 1

Omar E. Pol, Jul 14 2019

A001227(n) is denoted by Delta_0(n) in Glaisher 1907.

a(n) = A000203(n) iff n is a power of 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A000079, A000203, A001227, A062354, A062355, A064840.

Cf. A001620, A002117, A244115, A306016

Array[DivisorSum[#, 1 &, OddQ] DivisorSigma[1, #] &, 69] (* Michael De Vlieger, Nov 22 2019 *)
f[p_, e_] := (e+1)*(p^(e+1)-1)/(p-1); f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)

a(n) = sigma(n)*delta(n).
Multiplicative with a(2^e) = 2^(e+1) - 1 and a(p^e) = (e+1)*(p^(e+1)-1)/(p-1) for p > 2. - Amiram Eldar, Nov 01 2022
From Amiram Eldar, Dec 04 2023: (Start)
Dirichlet g.f.: (4^s - 3*2^s + 2)/(4^s - 2) * (zeta(s)*zeta(s-1))^2/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ (Pi^4/(168*zeta(3))) * n^2 * (log(n) + 2*gamma - 1/2 + 22*log(2)/21 + 2*zeta'(2)/zeta(2) - 2*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

A324986 a(n) = Sum_{d|n} (tau(d)*sigma(d)) where tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 7, 9, 28, 13, 63, 17, 88, 48, 91, 25, 252, 29, 119, 117, 243, 37, 336, 41, 364, 153, 175, 49, 792, 106, 203, 208, 476, 61, 819, 65, 621, 225, 259, 221, 1344, 77, 287, 261, 1144, 85, 1071, 89, 700, 624, 343, 97, 2187, 188, 742, 333, 812, 109, 1456, 325, 1496

Offset: 1

Jaroslav Krizek, Mar 23 2019

nonn

n divides a(n) for n: 1, 3, 4, 8, 12, 24, 28, 84, 88, 144, 264, 432, 440, 476, 1320, ...

Inverse Möbius transform of A064840. - Antti Karttunen, Mar 28 2019

			a(6) = tau(1)*sigma(1) + tau(2)*sigma(2) + tau(3)*sigma(3) + tau(6)*sigma(6) = (1*1) + (2*3) + (2*4) + (4*12) = 63.

Cf. A000005, A000203, A064840, A324987.

[&+[NumberOfDivisors(d) * SumOfDivisors(d): d in Divisors(n)]: n in [1..100]]

Table[Sum[DivisorSigma[0, k]*DivisorSigma[1, k], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Mar 23 2019 *)

a(n) = my(d=divisors(n)); sum(i=1, #d, numdiv(d[i])*sigma(d[i])) \\ Felix Fröhlich, Mar 23 2019

a(n) = sumdiv(n, d, numdiv(d)*sigma(d)); \\ Michel Marcus, Mar 24 2019

a(p) = 2p + 3 for p = primes (A000040).

a(n) = Sum_{d|n} A064840(d).

cp's OEIS Frontend

A064948 a(n) = Sum_{i|n, j|n} max(i,j).

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A071707 Numbers k that divide tau(k)*sigma(k).

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A233541 a(n) = sigma(n) + phi(n) + d(n).

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A328485 Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-1).

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A087801 Greatest common divisor of tau(n)+sigma(n) and tau(n)*sigma(n), where tau = A000005 and sigma = A000203.

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A126775 a(n) = phi(n)^2 * d(n) = A000010(n)^2 * A000005(n).

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A327830 Numbers m such that the geometric mean of tau(m) and sigma(m) is an integer.

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