A007429 -id:A007429 - cp's OEIS frontend

A280077 Partial sums of A007429 (Sum_{d|n} sigma(d)).

1, 5, 10, 21, 28, 48, 57, 83, 101, 129, 142, 197, 212, 248, 283, 340, 359, 431, 452, 529, 574, 626, 651, 781, 819, 879, 937, 1036, 1067, 1207, 1240, 1360, 1425, 1501, 1564, 1762, 1801, 1885, 1960, 2142, 2185, 2365, 2410, 2553, 2679, 2779, 2828, 3113, 3179

Offset: 1

Jaroslav Krizek, Dec 25 2016

nonn

sigma(n) is the sum of the divisors of n (A000203).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000203, A237349 (partial sums of A211776), A280078 (partial products of A007429).

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

a(n) = sum(k=1, n, sumdiv(k, d, sigma(d))); \\ Michel Marcus, May 29 2018

my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)*x^k/(1-x^k))/(1-x)) \\ Seiichi Manyama, Jul 24 2022

a(n) = Sum_{i=1..n} A007429(i).
a(n) = Sum_{k=1..n} A000203(k) * floor(n/k). - Daniel Suteu, May 28 2018
a(n) = Sum_{k=1..n} A000005(k)/2 * floor(n/k) * floor(1+n/k). - Daniel Suteu, May 28 2018
a(n) ~ Pi^4 * n^2 / 72. - Vaclav Kotesovec, Nov 06 2018
G.f.: (1/(1-x)) * Sum_{k>=1} sigma(k) * x^k/(1 - x^k). - Seiichi Manyama, Jul 24 2022

A280078 Partial products of A007429 (Sum_{d|n} sigma(d)).

Original entry on oeis.org

1, 4, 20, 220, 1540, 30800, 277200, 7207200, 129729600, 3632428800, 47221574400, 2597186592000, 38957798880000, 1402480759680000, 49086826588800000, 2797949115561600000, 53161033195670400000, 3827594390088268800000, 80379482191853644800000

Offset: 1

Jaroslav Krizek, Dec 25 2016

nonn

sigma(n) is the sum of the divisors of n (A000203).

Cf. A000203, A280075 (partial products of A211776), A280077 (partial sums of A007429).

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

a(n) = Product_{i=1..n} A007429(i).

A296075 Sum of deficiencies of divisors of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 7, 4, 8, 8, 11, 1, 13, 12, 13, 5, 17, 6, 19, 7, 19, 20, 23, -10, 24, 24, 22, 13, 29, 4, 31, 6, 31, 32, 33, -16, 37, 36, 37, -2, 41, 12, 43, 25, 30, 44, 47, -37, 48, 34, 49, 31, 53, 8, 53, 6, 55, 56, 59, -49, 61, 60, 46, 7, 63, 28, 67, 43, 67, 36, 71, -78, 73, 72, 58, 49, 75, 36, 79, -27, 63, 80, 83, -47, 83

Offset: 1

Antti Karttunen, Dec 04 2017

a(n)=0 for n in A066218. Are 1 and 12 the only solutions to a(n)=1? - Robert Israel, Dec 04 2017

			For n = 6, whose divisors are 1, 2, 3, 6, their deficiencies are 1, 1, 2, 0, thus a(6) = 1 + 1 + 2 + 0 = 4.
For n = 24, whose divisors are 1, 2, 3, 4, 6, 8, 12, 24, their deficiencies are 1, 1, 2, 1, 0, 1, -4, -12, thus a(24) = 1 + 1 + 2 + 1 + 0 + 1 + -4 + -12 = -10.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A033879, A066218, A296074.

Cf. also A007429, A187793, A187794, A187795.

f:= n -> add(2*t-numtheory:-sigma(t), t=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Dec 04 2017

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2; a[1] = 1; a[n_] := Module[{f = FactorInteger[n]}, 2 * Times @@ f1 @@@ f - Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Dec 04 2023 *)

A033879(n) = ((2*n)-sigma(n));
A296075(n) = sumdiv(n,d,A033879(d));

a(n) = Sum_{d|n} A033879(d).
a(n) = A296074(n) + A033879(n).
If m and n are coprime, a(m*n) = 2*a(m)*A000203(n)+2*a(n)*A000203(m)-a(m)*a(n)-2*A000203(m)*A000203(n). - Robert Israel, Dec 04 2017
a(n) = 2*A000203(n) - A007429(n). - Ridouane Oudra, Jul 29 2019
Sum_{k=1..n} a(k) ~ (Pi^2/6 - Pi^4/72) * n^2. - Amiram Eldar, Dec 04 2023

A321141 a(n) = Sum_{d|n} sigma_n(d).

Original entry on oeis.org

1, 6, 29, 291, 3127, 48246, 823545, 16909060, 387459858, 10019533302, 285311670613, 8920489178073, 302875106592255, 11113363271736486, 437893951444713443, 18447307036548136965, 827240261886336764179, 39346708467688595378892, 1978419655660313589123981

Offset: 1

Ilya Gutkovskiy, Oct 28 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386
N. J. A. Sloane, Transforms

Cf. A000005, A007429, A007433, A023887, A319194, A320940, A321140.

[&+[DivisorSigma(n, d):d in Divisors(n)]:n in [1..20]]; // Vincenzo Librandi, Feb 16 2020

with(numtheory): seq(coeff(series(add(sigma[n](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018

Table[Sum[DivisorSigma[n, d], {d, Divisors[n]}] , {n, 19}]
Table[SeriesCoefficient[Sum[DivisorSigma[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 19}]

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

from sympy import divisor_sigma, divisors
def A321141(n):
    return sum(divisor_sigma(d,0)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = [x^n] Sum_{k>=1} sigma_n(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d^n*tau(n/d).
a(n) ~ n^n. - Vaclav Kotesovec, Feb 16 2020

A066780 a(n) = Product_{k=1..n} sigma(k); sigma(k) is the sum of the positive divisors of n.

Original entry on oeis.org

1, 3, 12, 84, 504, 6048, 48384, 725760, 9434880, 169827840, 2037934080, 57062154240, 798870159360, 19172883824640, 460149211791360, 14264625565532160, 256763260179578880, 10013767147003576320, 200275342940071526400, 8411564403483004108800

Offset: 1

Benoit Cloitre and Leroy Quet, Jan 18 2002

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = A007429(gcd(i,j)) for 1 <= i,j <= n. - Enrique Pérez Herrero, Aug 12 2011

Paul D. Hanna, Table of n, a(n) for n = 1..300 (terms 1..100 from _Harry J. Smith_).
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49
Vaclav Kotesovec, Plot of (a(n)^(1/n))/n for n = 1..10^7
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20).

Cf. A000203, A001088, A066843, A294343, A345144.

with(numtheory):seq(mul(sigma(k),k=1..n), n=1..26); # Zerinvary Lajos, Jan 11 2009
with(numtheory):a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=a[n-1]*sigma(n) od: seq(a[n], n=0..18); # Zerinvary Lajos, Mar 21 2009

A066780[n_] := Product[DivisorSigma[1,i], {i,1,n}]; Array[A066780,20] (* Enrique Pérez Herrero, Aug 12 2011 *)
FoldList[Times,DivisorSigma[1,Range[20]]] (* Harvey P. Dale, Jan 29 2022 *)

{ p=1; for (n=1, 100, write("b066780.txt", n, " ", p*=sigma(n)) ) } \\ Harry J. Smith, Mar 25 2010

Lim_{n->infinity} (a(n)^(1/n)) / n = A345144 / exp(1) = 0.57447937538407152396420163967936309825692994713661226083669171312803511135... - Vaclav Kotesovec, Jun 09 2021

A107759 a(n) = (+2)UnitarySigma(n): if n = Product p_i^r_i then a(n) = Product (2 + p_i^r_i).

Original entry on oeis.org

1, 4, 5, 6, 7, 20, 9, 10, 11, 28, 13, 30, 15, 36, 35, 18, 19, 44, 21, 42, 45, 52, 25, 50, 27, 60, 29, 54, 31, 140, 33, 34, 65, 76, 63, 66, 39, 84, 75, 70, 43, 180, 45, 78, 77, 100, 49, 90, 51, 108, 95, 90, 55, 116, 91

Offset: 1

Yasutoshi Kohmoto, May 25 2005

			a(12) = (2+3)*(2+4) = 30.

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

Cf. A007429, A034448, A054862, A072691, A107758, A330594.

A107759 := proc(n) local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; mul( 2+op(1,p)^op(2,p), p=pf) ; end if; end proc:
seq(A107759(n),n=1..60) ; # R. J. Mathar, Jan 07 2011

a[1] = 1; a[n_] := Times @@ (2 + Power @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d), where usigma = A034448. - Ilya Gutkovskiy, Mar 27 2020

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 1/p^2 - 2/p^3) = A072691 * A330594 = 0.910438... . - Amiram Eldar, Nov 01 2022

A127170 Triangle read by rows: T(n,k) is the number of divisors of n that are divisible by k, with 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 3, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 3, 0, 2, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 4, 3, 2, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Column k lists the terms of A000005 interleaved with k - 1 zeros.

Eigensequence of the triangle = A007557; i.e., sequence A007557 shifts to the left upon multiplication by A127170. - Gary W. Adamson, Apr 27 2009

			First 10 rows of the triangle:
  1;
  2, 1;
  2, 0, 1;
  3, 2, 0, 1;
  2, 0, 0, 0, 1;
  4, 2, 2, 0, 0, 1;
  2, 0, 0, 0, 0, 0, 1;
  4, 3, 0, 2, 0, 0, 0, 1;
  3, 0, 2, 0, 0, 0, 0, 0, 1;
  4, 2, 0, 0, 2, 0, 0, 0, 0, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)

Row sums give A007425.

Cf. A000005, A007557, A051731, A126988, A007429, A127170, A195050.

T:= (n, k)-> `if`(irem(n, k)=0, numtheory[tau](n/k), 0):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Feb 16 2022

Table[Function[D, Table[Count[D, ?(Mod[#, k] == 0 &)], {k, n}]]@ Divisors[n], {n, 12}] // Flatten (* _Michael De Vlieger, Feb 16 2022 *)

tabl(nn) = {m = matrix(nn, nn, n, k, if ((n % k) == 0, 1, 0)); m = m^2; for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ Michel Marcus, Apr 01 2015

A007429(n) = Sum_{i=1..n} i*a(i).

T(n,k) = A000005(n/k), if k divides n, otherwise 0, with n >= 1 and 1 <= k <= n. - Omar E. Pol, Apr 01 2015

8 terms taken from Example section and then corrected in Data section by Omar E. Pol, Mar 30 2015
Extended beyond a(21) by Omar E. Pol, Apr 01 2015
New name (which was a comment dated Mar 30 2015) from Omar E. Pol, Feb 16 2022

A191161 Hypersigma(n), definition 2: sum of the divisors of n plus the recursive sum of the divisors of the proper divisors.

Original entry on oeis.org

1, 4, 5, 12, 7, 22, 9, 32, 19, 30, 13, 72, 15, 38, 37, 80, 19, 90, 21, 96, 47, 54, 25, 208, 39, 62, 65, 120, 31, 178, 33, 192, 67, 78, 65, 316, 39, 86, 77, 272, 43, 222, 45, 168, 147, 102, 49, 560, 67, 174, 97, 192, 55

Offset: 1

Alonso del Arte, May 26 2011

In wanting to ensure the definition was not arbitrary, I initially thought that 1s had to stop the recursion. But as T. D. Noe showed me, this doesn't have to be the case: the 1s can be included in the recursion.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A191161, Sequence Machine.

Cf. A000203, A191150, A202687, A255242, A378211 (Dirichlet inverse).

Sequences that appear in the convolution formulas: A000010, A000203, A007429, A038040, A060640, A067824, A074206, A174725, A253249, A323910, A323912, A330575.

hsTD[n_] := hsTD[n] = Module[{d = Divisors[n]}, Total[d] + Total[hsTD /@ Most[d]]]; Table[hsTD[n], {n, 100}] (* From T. D. Noe *)

a(n)=sumdiv(n,d,if(dCharles R Greathouse IV, Dec 20 2011

a(n) = sigma(n) + sum_{d | n, d < n} a(d). - Charles R Greathouse IV, Dec 20 2011
From Antti Karttunen, Nov 22 2024: (Start)
Following formulas were conjectured by Sequence Machine:
For n > 1, a(n) = A191150(n) + A074206(n).
a(n) = A330575(n) + A255242(n) = 2*A255242(n) + n = 2*A330575(n) - n.
a(n) = Sum_{d|n} A330575(d).
a(n) = Sum_{d|n} d*A067824(n/d).
a(n) = Sum_{d|n} A000203(d)*A074206(n/d).
a(n) = Sum_{d|n} A007429(d)*A174725(n/d).
a(n) = Sum_{d|n} A000010(d)*A253249(n/d).
a(n) = Sum_{d|n} A038040(d)*A323912(n/d).
a(n) = Sum_{d|n} A060640(d)*A323910(n/d).
(End)

A007430 Inverse Moebius transform applied thrice to natural numbers.

Original entry on oeis.org

1, 5, 6, 16, 8, 30, 10, 42, 24, 40, 14, 96, 16, 50, 48, 99, 20, 120, 22, 128, 60, 70, 26, 252, 46, 80, 82, 160, 32, 240, 34, 219, 84, 100, 80, 384, 40, 110, 96, 336, 44, 300, 46, 224, 192, 130, 50, 594, 76, 230, 120, 256, 56, 410, 112, 420, 132, 160, 62, 768, 64, 170, 240, 466, 128, 420

Offset: 1

N. J. A. Sloane

a(n) = A000027(n) * A000012(n) * A000012(n) * A000012(n) = A000027(n) * A000012(n) * A000005(n) = A000203(n) * A000005(n) = A000203(n) * A000012(n) * A000012(n) = A007429(n) * A000012(n), where operation * denotes Dirichlet convolution for n >= 1. Dirichlet convolution of functions b(n), c(n) is function a(n) = b(n) * c(n) = Sum_{d|n} b(d)*c(n/d). - Jaroslav Krizek, Mar 20 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
O. Bordelles, Mean values of generalized gcd-sum and lcm-sum functions, JIS 10 (2007) 07.9.2, sequence g_5.
N. J. A. Sloane, Transforms

Cf. A000005, A000203, A027750.

a007430 n = sum $ zipWith (*) (map a000005 ds) (map a000203 $ reverse ds)
            where ds = a027750_row n
-- Reinhard Zumkeller, Aug 02 2014

a[n_] := Total[ DivisorSigma[1, #]*DivisorSigma[0, n/#]& /@ Divisors[n]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 15 2011 *)

PARI
```
a(n)=sumdiv(n,d,sigma(d)*numdiv(n/d))
```

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3/(1-p*X))[n]) /* Ralf Stephan */

a(n)=sumdiv(n, x, sumdiv(x, y, sumdiv(y, z, z ) ) ); /* Joerg Arndt, Oct 07 2012 */

a(n) = Sum_{d|n} sigma(d)*tau(n/d). - Benoit Cloitre, Mar 03 2004
Multiplicative with a(p^e) = Sum_{k=0..e} binomial(e-k+2, e-k)*p^k.
Dirichlet g.f.: zeta(s-1)*zeta^3(s).
Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 432. - Vaclav Kotesovec, Nov 06 2018

A068978 Numbers k such that Sum_{d|k} tau(d)/d is an integer, where tau(x) = A000005(x).

Original entry on oeis.org

1, 2, 9, 18, 105, 210, 24375, 48750, 133848, 18780741, 18780965, 37561482, 37561930, 121486365, 169028685, 242972730, 338057370, 360988056, 676114740, 1120584213, 1285201500, 1352229480, 2241168426, 2776831200, 5352575025, 5408917920, 7437262140, 10705150050

Offset: 1

Benoit Cloitre, Apr 06 2002

nonn

Also k such that k divides A007429(k).
Also k such that k divides A211780(k). - Jaroslav Krizek, Sep 28 2014
a(28) > 10^10. - Giovanni Resta, Jun 10 2013
a(33) > 5*10^10. - Hiroaki Yamanouchi, Oct 05 2014

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..32
Hiroaki Yamanouchi, Conjectured table of n, a(n) for n = 1..200

Cf. A000005, A007429, A211780.

t = {}; n = 0; While[n++ <= 20000000, If[Mod[Total[DivisorSigma[1, Divisors[n]]], n] == 0, AppendTo[t, n]]]; t (* Jayanta Basu, Apr 03 2013 *)
f[p_, e_] := (p*(p^(e+1) - 1) - (p-1)*(e+1))/(p-1)^2; q[1] = True; q[k_] := Divisible[Times @@ f @@@ FactorInteger[k], k]; Select[Range[200000], q] (* Amiram Eldar, Apr 19 2025 *)

for(n=1, 20000000, if(denominator( sumdiv(n,d, numdiv(d)/d)) ==1, print1(n,",")))

isok(k) = {my(f = factor(k)); !(prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p*(p^(e+1) - 1) - (p-1)*(e+1))/(p-1)^2) % k);} \\ Amiram Eldar, Apr 19 2025

More terms from Rick L. Shepherd, Jun 23 2002
a(12)-a(27) from Giovanni Resta, Jun 10 2013
a(28) from Hiroaki Yamanouchi, Oct 05 2014

cp's OEIS Frontend

A280077 Partial sums of A007429 (Sum_{d|n} sigma(d)).

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A280078 Partial products of A007429 (Sum_{d|n} sigma(d)).

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A296075 Sum of deficiencies of divisors of n.

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A321141 a(n) = Sum_{d|n} sigma_n(d).

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A066780 a(n) = Product_{k=1..n} sigma(k); sigma(k) is the sum of the positive divisors of n.

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A107759 a(n) = (+2)UnitarySigma(n): if n = Product p_i^r_i then a(n) = Product (2 + p_i^r_i).

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A127170 Triangle read by rows: T(n,k) is the number of divisors of n that are divisible by k, with 1 <= k <= n.

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