A007429 -id:A007429 - cp's OEIS frontend

A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

1, 3, 4, 11, 6, 42, 8, 75, 31, 108, 12, 1778, 14, 206, 234, 1099, 18, 5901, 20, 8116, 452, 498, 24, 333618, 131, 692, 760, 22166, 30, 810372, 32, 33867, 1104, 1176, 1238, 10085333, 38, 1466, 1538, 2568180, 42, 3112382, 44, 85690, 91386, 2142, 48, 255138610

Offset: 1

Jaroslav Krizek, Apr 01 2010

nonn

			For n = 4, with b(n) = A007955(n), a(4) = b(1) + b(2) + b(4) = 1 + 2 + 8 = 11.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007429, A007955, A206032, A266265.

Subsequences: A008864, A181388 \ {0}.

a[n_] := DivisorSum[n, #^(DivisorSigma[0, #]/2) &]; Array[a, 50] (* Amiram Eldar, Oct 23 2021 *)

a(n) = sumdiv(n, d, vecprod(divisors(d))); \\ Michel Marcus, Dec 09 2014 and Oct 23 2021

from math import isqrt
from sympy import divisor_count, divisors
def A175317(n): return sum(isqrt(d)**c if (c:=divisor_count(d)) & 1 else d**(c//2) for d in divisors(n,generator=True)) # Chai Wah Wu, Jun 24 2022

From Bernard Schott, Oct 26 2021: (Start)
a(1) = 1 (the only fixed point).
a(p) = p+1 for prime p only.
a(2^k) = A181388(k+1). (End)

Corrected by Jaroslav Krizek, Apr 02 2010

Edited and more terms from Michel Marcus, Dec 09 2014

A211779 a(n) = Sum_{d_A000203(x).

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, 70, 7, 18, 18, 43, 1, 68, 1, 57, 17, 22, 15, 107, 1, 24, 19, 92, 1, 84, 1, 59, 48, 28, 1, 161, 9, 59, 23, 67, 1, 112, 19, 114, 25, 34, 1, 217, 1, 36, 58, 120, 21, 116, 1, 83, 29

Offset: 1

Jaroslav Krizek, Apr 20 2012

nonn

The numbers n < 1000 such that n divides a(n) are 4, 10, 42, and 90. (See A224488).

Antti Karttunen, Table of n, a(n) for n = 1..27144 (first 1000 terms from Jaroslav Krizek)
Index entries for sequences related to sums of divisors.

Cf. A000203, A007429, A001065, A211780, A224488.

Table[Sum[DivisorSigma[1, d], {d, Most[Divisors[n]]}], {n, 100}] (* T. D. Noe, Apr 26 2012 *)

a(n)=sumdiv(n,d,sigma(d))-sigma(n) \\ Charles R Greathouse IV, Feb 19 2013

a(n) = A007429(n) - A000203(n) = A211780(n) - A000203(n) + n.
G.f.: sum(n>=1, A000203(n)*x^(2*n)/(1-x^n) ). - Mircea Merca, Feb 26 2014
a(n) = Sum_{d|n} A001065(d). - Antti Karttunen, Nov 13 2017
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 - Pi^2/12 = 0.530437... . - Amiram Eldar, Mar 17 2024

A263832 The number c_{Cc,pi_1(B_2)}(n) of the second amphicosm n-coverings over the second amphicosm.

Original entry on oeis.org

1, 0, 5, 2, 7, 0, 9, 6, 18, 0, 13, 10, 15, 0, 35, 14, 19, 0, 21, 14, 45, 0, 25, 30, 38, 0, 58, 18, 31, 0, 33, 30, 65, 0, 63, 36, 39, 0, 75, 42, 43, 0, 45, 26, 126, 0, 49, 70, 66, 0, 95, 30, 55, 0, 91, 54, 105, 0, 61, 70, 63, 0, 162, 62, 105, 0, 69

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..20000
G. Chelnokov, M. Deryagina, A. Mednykh, On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2, arXiv preprint arXiv:1502.01528 [math.AT], 2015.
G. Chelnokov, M. Deryagina and A. Mednykh, On the coverings of Euclidean manifolds B_1 and B_2, Communications in Algebra, Vol. 45, No. 4 (2017), 1558-1576.

Cf. A263825, A263826, A263827, A263828, A263829, A263830, A263831.

sigma[n_] := DivisorSigma[1, n]; q = Quotient;
a[n_] := Switch[Mod[n, 4], 0, Sum[sigma[q[n, 2d]] - sigma[q[n, 4d]], {d, Divisors[q[n, 4]]}], 2, 0, 1|3, Sum[sigma[d], {d, Divisors[n]}]];
Array[a, 70] (* Jean-François Alcover, Dec 01 2018, after Gheorghe Coserea *)

A007429(n) = sumdiv(n, d, sigma(d));
a(n) = {
  if (n%2, A007429(n), if (n%4, 0,
      sumdiv(n\4, d, sigma(n\(2*d)) - sigma(n\(4*d)))));
};
vector(67, n, a(n))  \\ Gheorghe Coserea, May 05 2016

More terms from Gheorghe Coserea, May 05 2016

A319131 a(n) = Sum_{d|n} Sum_{p|d, p prime} p.

Original entry on oeis.org

0, 2, 3, 4, 5, 10, 7, 6, 6, 14, 11, 17, 13, 18, 16, 8, 17, 18, 19, 23, 20, 26, 23, 24, 10, 30, 9, 29, 29, 40, 31, 10, 28, 38, 24, 30, 37, 42, 32, 32, 41, 48, 43, 41, 27, 50, 47, 31, 14, 26, 40, 47, 53, 26, 32, 40, 44, 62, 59, 64, 61, 66, 33, 12, 36, 64, 67, 59, 52, 56

Offset: 1

Ilya Gutkovskiy, Sep 11 2018

Inverse Möbius transform of A008472.

			a(12) = 13 as 12 has 6 divisors and 2 * 6 * (2/3) + 3 * 6 * (1/2) = 17. - _David A. Corneth_, Oct 08 2019

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

Cf. A000005, A007429, A008472, A061397, A062799, A319132.

[0] cat  [&+[&+[PrimeDivisors(d)[i]:i in [1..#PrimeDivisors(d)]]:d in Set(Divisors(n)) diff {1}]:n in [2..70]]; // Marius A. Burtea, Oct 08 2019

[0] cat [&+[p*#Divisors(n div p):p in PrimeDivisors(n)]:n in [2..70]]; // Marius A. Burtea, Oct 08 2019 (According to the formula given by Ridouane Oudra)

with(numtheory): seq(add(p*tau(n/p), p in factorset(n)), n=1..80); # Ridouane Oudra, Oct 08 2019

Table[Sum[Total[Select[Divisors[d], PrimeQ]], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, PrimeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, PrimeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

a(n) = sumdiv(n, d, my(f=factor(d)); vecsum(f[,1])); \\ Michel Marcus, Oct 08 2019

a(n) = my(f = factor(n), nd = numdiv(f)); sum(i = 1, #f~, f[i, 1] * nd / (f[i, 2] + 1) * f[i, 2]) \\ David A. Corneth, Oct 08 2019

G.f.: Sum_{k>=1} A008472(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A008472(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(p^k) = p*k, where p is a prime.
a(n) = Sum_{p|n} p*tau(n/p), where p is a prime and tau(n) = A000005(n). - Ridouane Oudra, Oct 08 2019
a(n) = Sum_{p|n} p*tau(n)*(e_p-1)/(e_p) where e_p is the exponent of p in the factorization of n. - David A. Corneth, Oct 08 2019
a(n) = Sum_{d|n} sopf(d). - Wesley Ivan Hurt, May 23 2021

A134577 A127170 * A127648.

Original entry on oeis.org

1, 2, 2, 2, 0, 3, 3, 4, 0, 4, 2, 0, 0, 0, 5, 4, 4, 6, 0, 0, 6, 2, 0, 0, 0, 0, 0, 7, 4, 6, 0, 8, 0, 0, 0, 8, 3, 0, 6, 0, 0, 0, 0, 0, 9, 4, 4, 0, 0, 10, 0, 0, 0, 0, 10

Offset: 1

Gary W. Adamson, Nov 02 2007

Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, 26, ...).
Left border = A000005: (1, 2, 2, 3, 2, 4, 2, ...).
A134577 * [1/1, 1/2, 1/3, ...] = A007425: (1, 3, 3, 6, 3, 9, 3, 10, ...).
A134577 * [1, 2, 3, ...] = A007433: (1, 6, 11, 27, 27, 66, ...).
A134577 * A000005 = A034761: (1, 6, 8, 23, 12, 48, ...).

			First few rows of the triangle:
  1;
  2, 2;
  2, 0, 3;
  3, 4, 0, 4;
  2, 0, 0, 0, 5
  4, 4, 6, 0, 0, 6;
  2, 0, 0, 0, 0, 0, 7;
  4, 6, 0, 8, 0, 0, 0, 8;
  ...

Cf. A051731, A127170, A127648, A127093, A007429, A127170, A007433, A034761.

A127170 * A127648 = A051731^2 * A127648 = A051731 * A127093.

A221219 Numbers k such that sigma(k) divides Sum_{d|k} sigma(d).

Original entry on oeis.org

1, 198, 608, 4680, 11322, 20826, 56608, 60192, 179424, 1737000, 2578968, 3055150, 3441888, 5604192, 6008184, 6331104, 302459850, 320457888, 477229032, 565344850, 579667086, 589459104, 731925000, 766073448, 907521650, 928765600, 3586977576, 3732082848, 6487717600

Offset: 1

Paolo P. Lava, Feb 22 2013

nonn

A066218 is a subsequence of this sequence.
Numbers k such that A000203(k) divides A007429(k). - Jaroslav Krizek, Dec 22 2018
Corresponding values of (Sum_{d|k} sigma(d)) / sigma(k)  for numbers k from this sequence: 1, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, ... - Jaroslav Krizek, Dec 22 2018

			4680 is in the sequence because sigma(4680)=16380, its proper divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 18, 20, 24, 26, 30, 36, 39, 40, 45, 52, 60, 65, 72, 78, 90, 104, 117, 120, 130, 156, 180, 195, 234, 260, 312, 360, 390, 468, 520, 585, 780, 936, 1170, 1560, 2340 and the sum of their sigma values is 32760. Finally 32760/16380=2.

Cf. A000203, A066218, A224488, A322655, A322656, A319296.

[k: k in [1..1000000] | &+[SumOfDivisors(d): d in Divisors(k)] mod  SumOfDivisors(k) eq 0] // Jaroslav Krizek, Dec 22 2018

with(numtheory);
A221219:=proc(q) local a,b,j,n;
for n from 1 to q do a:=divisors(n); b:=add(sigma(a[j]),j=1..nops(a));
  if type(b/sigma(n),integer) then print(n); fi; od; end:
A221219(10^10);

f1[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f2[p_, e_] := (p^(e+1) - 1)/(p - 1); aQ[1] = True; aQ[n_] := Module[{f = FactorInteger[n]}, Divisible[Times @@ f1 @@@ f, Times @@ f2 @@@ f]]; Select[Range[10^5], aQ] (* Amiram Eldar, Dec 23 2018 *)

isok(n) = (sumdiv(n, d, sigma(d)) % sigma(n) == 0); \\ Michel Marcus, Dec 22 2018

a(10)-a(28) from Donovan Johnson, Apr 05 2013

1 prepended by Jaroslav Krizek, Dec 22 2018

A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 1, 5, 23, 179, 1279, 13699, 135085, 1764377, 22527521, 344625461, 5283739471, 94562354875, 1685808248383, 33947023942259, 694786150879829, 15613612524749489, 357353282848083265, 8880505496901812197, 224851013929747732231, 6106205671049245677251, 169523515381173773551871

Offset: 0

Ilya Gutkovskiy, May 26 2018

nonn

a(n)/n! is the Euler transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Vaclav Kotesovec, Table of n, a(n) for n = 0..438
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A000203, A007429, A017665, A017666, A028342, A053529, A061256, A072169, A318769, A318814.

with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018

A318491 a(n) is the numerator of Sum_{d|n} Sum_{j|d} 1/j.

Original entry on oeis.org

1, 5, 7, 17, 11, 35, 15, 49, 34, 11, 23, 119, 27, 75, 77, 129, 35, 85, 39, 187, 5, 115, 47, 343, 86, 135, 142, 255, 59, 77, 63, 321, 161, 175, 33, 289, 75, 195, 63, 539, 83, 25, 87, 391, 374, 235, 95, 301, 162, 43, 245, 459, 107, 355, 23, 105, 91, 295, 119, 1309, 123, 315, 170, 769, 297

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, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A318492 (denominators).

Cf. A001620, A013661, A306016.

f:= proc(n) local d;
   numer(add(numtheory:-sigma(d)/d, d = numtheory:-divisors(n))) end proc:
map(f, [$1..65]); # Robert Israel, Jan 13 2025

Numerator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 65}]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]

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

Numerators 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).
Numerators 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) = numerator of Sum_{d|n} sigma(d)/d.
a(n) = numerator of (1/n)*Sum_{d|n} d*tau(d).
If p is a prime, a(p) = 2*p + 1.
Sum_{k=1..n} a(k)/A318492(k) ~ zeta(2) * n * (log(n) + 2*gamma - 1 + zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2024

A344044 a(n) = Sum_{d|n} sigma(d)^3.

Original entry on oeis.org

1, 28, 65, 371, 217, 1820, 513, 3746, 2262, 6076, 1729, 24115, 2745, 14364, 14105, 33537, 5833, 63336, 8001, 80507, 33345, 48412, 13825, 243490, 30008, 76860, 66262, 190323, 27001, 394940, 32769, 283584, 112385, 163324, 111321, 839202, 54873, 224028, 178425, 812882, 74089

Offset: 1

Seiichi Manyama, May 08 2021

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

Cf. A002117, A007429, A065018, A344043, A344047.

a[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Array[a, 41] (* Amiram Eldar, May 08 2021 *)

PARI
```
a(n) = sumdiv(n, d, sigma(d)^3);
```

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k)^3*x^k/(1-x^k)))

G.f.: Sum_{k >= 1} sigma(k)^3 * x^k/(1 - x^k).
If p is prime, a(p) = 1 + (p+1)^3.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (Pi^10*zeta(3)/194400) * Product_{p prime} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 1.6422194986... . - Amiram Eldar, Nov 20 2022

A127172 Cube of A051731.

Original entry on oeis.org

1, 3, 1, 3, 0, 1, 6, 3, 0, 1, 3, 0, 0, 0, 1, 9, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 10, 6, 0, 3, 0, 0, 0, 1, 6, 0, 3, 0, 0, 0, 0, 0, 1, 9, 3, 0, 0, 0, 3, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Nonzero terms in every column = A007425: (1, 3, 3, 6, 3, 9, 3, ...).
Row sums = A007426: (1, 4, 4, 20, 4, 16, ...).
A127172 * mu(n) = d(n); or A127172 * A008683 = A000005.
A127172 * d(n) = tau_5(n); or A127172 * A000005 = A061200.
A127172 * phi(n) = A007429: (1, 4, 5, 11, 7, 20, ...); or: A127172 * A000010 = A007429.
Note that A051731 * d(n) = row sums of A127172; or A051731 * A000005 = A007425.
Also, A126988 * mu(n) = phi(n); or A126988 * A008683 = A000010.
A126988 * phi(n) = A018804: (1, 3, 5, 8, 9, 15, ...); = A127170 * mu(n).

			First few rows of the triangle:
   1;
   3, 1;
   3, 0, 1;
   6, 3, 0, 1;
   3, 0, 0, 0, 1;
   9, 3, 3, 0, 0, 1;
   3, 0, 0, 0, 0, 0, 1;
  10, 6, 0, 3, 0, 0, 0, 1;
   6, 0, 3, 0, 0, 0, 0, 0, 1;
   9, 3, 0, 0, 3, 0, 0, 0, 0, 1;
  ...

Cf. A000005, A007425, A127170, A051731, A007429, A000010, A126988, A018804.

Cube of A051731 A007425: (1, 3, 3, 6, 3, 9, 3, ...) in every column k, interspersed with (k-1) zeros.

cp's OEIS Frontend

A175317 a(n) = Sum_{d|n} A007955(d) where A007955(m) = product of divisors of m.

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A211779 a(n) = Sum_{d_A000203(x).

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A263832 The number c_{Cc,pi_1(B_2)}(n) of the second amphicosm n-coverings over the second amphicosm.

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A319131 a(n) = Sum_{d|n} Sum_{p|d, p prime} p.

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A134577 A127170 * A127648.

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A221219 Numbers k such that sigma(k) divides Sum_{d|k} sigma(d).

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Magma

Maple

Mathematica

PARI

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A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

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