A007429 -id:A007429 - cp's OEIS frontend

A318492 a(n) is the denominator of Sum_{d|n} Sum_{j|d} 1/j.

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 11, 12, 13, 14, 15, 16, 17, 9, 19, 20, 1, 22, 23, 24, 25, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 18, 37, 38, 13, 40, 41, 2, 43, 44, 45, 46, 47, 16, 49, 5, 51, 52, 53, 27, 5, 8, 19, 58, 59, 60, 61, 62, 21, 64, 65, 66, 67, 4, 69, 14, 71, 36, 73, 74, 75

Offset: 1

Ilya Gutkovskiy, Aug 27 2018

			1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A068986 (positions of 1's), A318491 (numerators).

Denominator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 75}]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]

a(n) = denominator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Denominators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).
Denominators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).
a(n) = denominator of Sum_{d|n} sigma(d)/d.
a(n) = denominator of (1/n)*Sum_{d|n} d*tau(d).

A319132 a(n) = Sum_{d|n} Sum_{j|d} mu(j)^2*j, where mu = Möbius function (A008683).

Original entry on oeis.org

1, 4, 5, 7, 7, 20, 9, 10, 9, 28, 13, 35, 15, 36, 35, 13, 19, 36, 21, 49, 45, 52, 25, 50, 13, 60, 13, 63, 31, 140, 33, 16, 65, 76, 63, 63, 39, 84, 75, 70, 43, 180, 45, 91, 63, 100, 49, 65, 17, 52, 95, 105, 55, 52, 91, 90, 105, 124, 61, 245, 63, 132, 81, 19, 105, 260, 69, 133, 125, 252

Offset: 1

Ilya Gutkovskiy, Sep 11 2018

Inverse Möbius transform of A048250.

Metin Sariyar, Table of n, a(n) for n = 1..16000
N. J. A. Sloane, Transforms.

Cf. A007429, A008683, A048250, A048691, A055615, A072691, A319131.

[&+[&+[MoebiusMu(j)^2*j:j in Divisors(d)]:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019

[&+[MoebiusMu(d)^2*d*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019

with(numtheory): seq(add(mobius(d)^2*d*tau(n/d), d in divisors(n)), n=1..70); # Ridouane Oudra, Nov 13 2019

Table[Sum[Sum[MoebiusMu[j]^2 j, {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, SquareFreeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (p + 1)*e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = sumdiv(n, d, moebius(d)^2*d*numdiv(n/d)); \\ Michel Marcus, Nov 13 2019; corrected Jun 13 2022

G.f.: Sum_{k>=1} A048250(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A048250(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(p^k) = (p + 1)*k + 1, where p is a prime.
a(n) = Sum_{d|n} mu(d)^2*d*tau(n/d). - Ridouane Oudra, Nov 13 2019
Multiplicative with a(p^e) = (p+1)*e+1. - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 = 0.822467... (A072691). - Amiram Eldar, Nov 13 2022
Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2*s-2). - Amiram Eldar, Jan 03 2023

A319296 a(n) = (Sum_{d|n} sigma(d)) mod sigma(n).

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 11, 5, 10, 1, 27, 1, 12, 11, 26, 1, 33, 1, 35, 13, 16, 1, 10, 7, 18, 18, 43, 1, 68, 1, 57, 17, 22, 15, 16, 1, 24, 19, 2, 1, 84, 1, 59, 48, 28, 1, 37, 9, 59, 23, 67, 1, 112, 19, 114, 25, 34, 1, 49, 1, 36, 58, 120, 21, 116, 1, 83, 29, 108

Offset: 1

Jaroslav Krizek, Sep 16 2018

nonn

			For n = 4; a(4) = (sigma(1) + sigma(2) + sigma(4)) mod sigma(4) = (1+3+7) mod 7 = 11 mod 7 = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000040, A000203, A007429, A221219.

[&+[SumOfDivisors(d): d in Divisors(n)] mod  SumOfDivisors(n): n in [1..1000]];

Table[Mod[Sum[DivisorSigma[1, d], {d, Divisors[n]}], DivisorSigma[1, n]], {n, 1, 100}] (* Vaclav Kotesovec, Sep 26 2018 *)

A319296(n) = (sumdiv(n,d,sigma(d))%sigma(n)); \\ Antti Karttunen, Sep 16 2018

a(n) = A007429(n) mod A000203(n).
a(A221219(n)) = 0.
a(A000040(n)) = 1; the only composite number < 2*10^6 with a(n) = 1 is 636.
a(n) = n only for numbers 4, 10 and 96 < 3000000.

A321140 a(n) = Sum_{d|n} sigma_3(d).

Original entry on oeis.org

1, 10, 29, 83, 127, 290, 345, 668, 786, 1270, 1333, 2407, 2199, 3450, 3683, 5349, 4915, 7860, 6861, 10541, 10005, 13330, 12169, 19372, 15878, 21990, 21226, 28635, 24391, 36830, 29793, 42798, 38657, 49150, 43815, 65238, 50655, 68610, 63771, 84836, 68923, 100050, 79509, 110639, 99822

Offset: 1

Ilya Gutkovskiy, Oct 28 2018

Inverse Möbius transform applied twice to cubes.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A001158, A007429, A007433, A027848.

with(numtheory): seq(coeff(series(add(sigma[3](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 28 2018

Table[Sum[DivisorSigma[3, d], {d, Divisors[n]}] , {n, 45}]
nmax = 45; Rest[CoefficientList[Series[Sum[DivisorSigma[3, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
f[p_, e_] := (p^3*(p^(3e+3) - e - 2) + e + 1)/(p^3 - 1)^2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = sumdiv(n, d, sigma(d, 3)); \\ Michel Marcus, Oct 28 2018

G.f.: Sum_{k>=1} sigma_3(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d^3*tau(n/d).
From Jianing Song, Oct 28 2018: (Start)
Multiplicative with a(p^e) = (p^3*(p^(3e+3) - e - 2) + e + 1)/(p^3 - 1)^2.
Dirichlet g.f.: zeta(s)^2*zeta(s-3). (End)
Sum_{k=1..n} a(k) ~ Pi^8 * n^4 / 32400. - Vaclav Kotesovec, Nov 08 2018

A145398 a(n) = Sum_{d|n} sigma(d) - Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).

Original entry on oeis.org

1, 3, 5, 11, 7, 15, 9, 31, 18, 21, 13, 55, 15, 27, 35, 75, 19, 54, 21, 77, 45, 39, 25, 155, 38, 45, 58, 99, 31, 105, 33, 167, 65, 57, 63, 198, 39, 63, 75, 217, 43, 135, 45, 143, 126, 75, 49, 375, 66, 114, 95, 165, 55, 174, 91, 279, 105, 93, 61, 385, 63, 99, 162, 355, 105, 195

Offset: 1

N. J. A. Sloane, Mar 13 2009

Dirichlet convolution of [1,-1,0,4,0,0,...] with A007429.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. Table 1, symmetry C2/m.

Cf. A007429, A145378, A152649.

read("transforms") ;  s1 := [1,-1,0,4,seq(0,n=1..40)] ; s2 := [seq(add(sigma(d),d=divisors(n)),n=1..40)] ; DIRICHLET(s1,s2) ; # R. J. Mathar, Feb 07 2011

f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 3*2^(e + 1) - 4*e - 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*2^(f[i,2]+1) - 4*f[i,2] - 5,  (f[i,1]*(f[i,1]^(f[i,2]+1)-1) - (f[i,1]-1)*(f[i,2]+1))/(f[i,1]-1)^2)); } \\ Amiram Eldar, Oct 25 2022

Dirichlet g.f.: (1-1/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).
From Amiram Eldar, Oct 25 2022: (Start):
Multiplicative with a(2^e) = 3*2^(e+1)-4*e-5, and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). (End)

A211780 a(n) = Sum_{d|n, dA000005 is the number of divisors.

Original entry on oeis.org

0, 2, 2, 7, 2, 14, 2, 18, 9, 18, 2, 43, 2, 22, 20, 41, 2, 54, 2, 57, 24, 30, 2, 106, 13, 34, 31, 71, 2, 110, 2, 88, 32, 42, 28, 162, 2, 46, 36, 142, 2, 138, 2, 99, 81, 54, 2, 237, 17, 102, 44, 113, 2, 178, 36, 178, 48, 66, 2, 325, 2, 70, 99, 183, 40, 194, 2

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

Numbers n such that n divides a(n) are given in A068978.

			For n = 12: Sum_{d|n, d<n} d * tau(n / d) = 1*6 + 2*4 + 3*3 + 4*2 + 6*2 = 43.

Antti Karttunen, Table of n, a(n) for n = 1..27144 (first 1000 terms from Jaroslav Krizek)

Cf. A000005, A000203, A007429, A068978, A098198, A211779.

Table[Sum[d*DivisorSigma[0, n/d], {d, Most[Divisors[n]]}], {n, 100}] (* T. D. Noe, Apr 27 2012 *)

A211780(n) = sumdiv(n, d, sigma(d))-n; \\ Antti Karttunen, Nov 13 2017

A211780=lambda n:sum(sigma(d) for d in divisors(n, generator=True))-n
from sympy import divisor_sigma as sigma, divisors # M. F. Hasler, Jun 03 2024

a(n) = A007429(n) - n = A211779(n) + A000203(n) - n .
a(n) = (Sum_{d|n} A000203(d)) - n. - Antti Karttunen, Nov 13 2017
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^4/36 - 1 = 1.705808... . - Amiram Eldar, Jun 06 2024

Name edited by M. F. Hasler, Jun 03 2024

A276736 a(n) = numerator of Sum_{d|n} tau(d)/d.

Original entry on oeis.org

1, 2, 5, 11, 7, 10, 9, 13, 2, 14, 13, 55, 15, 18, 7, 57, 19, 4, 21, 77, 15, 26, 25, 65, 38, 30, 58, 99, 31, 14, 33, 15, 65, 38, 9, 11, 39, 42, 25, 91, 43, 30, 45, 13, 14, 50, 49, 95, 66, 76, 95, 165, 55, 116, 91, 117, 35, 62, 61, 77, 63, 66, 18, 247, 21, 130

Offset: 1

Jaroslav Krizek, Sep 16 2016

Also numerators of (Sum_{d|n} sigma(d)) / n.

			For n=6; {d_6} = {1, 2, 3, 6}; {tau(d)_6} = {1, 2, 2, 4};  Sum_{d|6} tau(d)/d = 1/1 + 2/2 + 2/3 + 4/6 = 20/6 = 10/3; a(6) = 10.
For n=6; {d_6} = {1, 2, 3, 6}; {sigma(d)_6} = {1, 3, 4, 12};  (Sum_{d|6} sigma(d))/6 = (1+3+4+12)/6 = 10/3; a(6) = 10.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000005, A007429, A276737 (denominators).

[Numerator(&+[NumberOfDivisors(d)/d: d in Divisors(n)]): n in [1..100]]

Table[Numerator@ Total[DivisorSigma[0, #]/#] &@ Divisors@ n, {n, 66}] (* Michael De Vlieger, Sep 16 2016 *)

a(n) = numerator(sumdiv(n, d, numdiv(d)/d)); \\ Michel Marcus, Sep 16 2016

For all n we have: n = (Sum_{d|n} sigma(d)) / (Sum_{d|n} tau(d)/d) = (Sum_{d|n} d*tau(n/d)) / (Sum_{d|n} tau(d)/d) = A007429(n) * A276737(n) / a(n).

A276737 a(n) = denominator of Sum_{d|n} tau(d)/d.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 4, 1, 5, 11, 12, 13, 7, 3, 16, 17, 1, 19, 20, 7, 11, 23, 12, 25, 13, 27, 28, 29, 3, 31, 4, 33, 17, 5, 2, 37, 19, 13, 20, 41, 7, 43, 4, 5, 23, 47, 16, 49, 25, 51, 52, 53, 27, 55, 28, 19, 29, 59, 12, 61, 31, 7, 64, 13, 33, 67, 68, 69, 5, 71

Offset: 1

Jaroslav Krizek, Sep 16 2016

Also denominator of (Sum_{d|n} sigma(d)) / n.

			For n=6; {d_6} = {1, 2, 3, 6}; {tau(d)_6} = {1, 2, 2, 4}; Sum_{d|6} tau(d)/d = 1/1 + 2/2 + 2/3 + 4/6 = 20/6 = 10/3; a(6) = 3.
For n=6; {d_6} = {1, 2, 3, 6}; {sigma(d)_6} = {1, 3, 4, 12};  (Sum_{d|6} sigma(d))/6 = (1+3+4+12)/6 = 10/3; a(6) = 3.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000005, A007429, A068978, A276736 (numerators).

[Denominator(&+[NumberOfDivisors(d)/d: d in Divisors(n)]): n in [1..100]]

Table[Denominator@ Total[DivisorSigma[0, #]/#] &@ Divisors@ n, {n, 71}] (* Michael De Vlieger, Sep 16 2016 *)

a(n) = denominator(sumdiv(n, d, numdiv(d)/d)); \\ Michel Marcus, Sep 16 2016

a(A068978(n)) = 1.

For all n, n = (Sum_{d|n} sigma(d)) / (Sum_{d|n} tau(d)/d) = (Sum_{d|n} d*tau(n/d)) / (Sum_{d|n} tau(d)/d) = A007429(n) * a(n) / A276736(n).

A318446 Inverse Möbius transform of A005187: a(n) = Sum_{d|n} A005187(d).

Original entry on oeis.org

1, 4, 5, 11, 9, 18, 12, 26, 21, 30, 20, 47, 24, 40, 39, 57, 33, 68, 36, 75, 55, 64, 43, 108, 56, 76, 71, 100, 55, 126, 58, 120, 88, 102, 87, 167, 72, 112, 102, 168, 80, 174, 83, 156, 141, 134, 90, 233, 107, 174, 135, 184, 103, 222, 133, 224, 150, 170, 114, 309, 118, 180, 191, 247, 160, 272, 132, 243, 182, 270, 139, 370, 144

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005187, A318445.

Cf. also A297111, A300244.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A318446(n) = sumdiv(n,d,A005187(d));

a(n) = Sum_{d|n} A005187(d).
a(n) = A005187(n) + A318445(n).
a(n) = A318448(n) + A007429(n).

A318448 a(n) = Sum_{d|n} A294898(d), where A294898(d) = A005187(d) - sigma(d).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 2, 7, -8, 9, 4, 4, 0, 14, -4, 15, -2, 10, 12, 18, -22, 18, 16, 13, 1, 24, -14, 25, 0, 23, 26, 24, -31, 33, 28, 27, -14, 37, -6, 38, 13, 15, 34, 41, -52, 41, 22, 40, 19, 48, -10, 42, -10, 45, 46, 53, -76, 55, 48, 29, 0, 55, 12, 63, 34, 57, 18, 66, -98, 69, 64, 42, 37, 64, 16, 73, -42, 51, 72, 78, -74, 74, 74, 73, 6

Offset: 1

Antti Karttunen, Aug 27 2018

sign

Inverse Möbius transform of A294898.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A005187, A007429, A093653, A294898, A296075, A318446, A318447.

Cf. also A297114, A317844.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A318448(n) = sumdiv(n,d,A294898(d));

a(n) = Sum_{d|n} A294898(d).
a(n) = A318447(n) + A294898(n).
a(n) = A318446(n) - A007429(n).
a(n) = A296075(n) - A093653(n).

cp's OEIS Frontend

A318492 a(n) is the denominator of Sum_{d|n} Sum_{j|d} 1/j.

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A319132 a(n) = Sum_{d|n} Sum_{j|d} mu(j)^2*j, where mu = Möbius function (A008683).

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A319296 a(n) = (Sum_{d|n} sigma(d)) mod sigma(n).

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A321140 a(n) = Sum_{d|n} sigma_3(d).

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A145398 a(n) = Sum_{d|n} sigma(d) - Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).

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A211780 a(n) = Sum_{d|n, dA000005 is the number of divisors.

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A276736 a(n) = numerator of Sum_{d|n} tau(d)/d.

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