A007429 -id:A007429 - cp's OEIS frontend

A263827 The number c_{Cc pi_1(B_1)}(2n) of the second amphicosm 2n-coverings over the first amphicosm.

2, 6, 10, 14, 14, 30, 18, 30, 36, 42, 26, 70, 30, 54, 70, 62, 38, 108, 42, 98, 90, 78, 50, 150, 76, 90, 116, 126, 62, 210, 66, 126, 130, 114, 126, 252, 78, 126, 150, 210, 86, 270, 90, 182, 252, 150, 98, 310, 132, 228, 190, 210, 110, 348, 182, 270, 210, 186, 122, 490, 126, 198, 324, 254, 210, 390, 138, 266, 250, 378

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.

Cf. A263825-A263830, A263832.

A263827 := proc(n)
    local locn,a,twol,fourl ;
    locn := 2*n ;
    # Theorem 3 (iii)
    a := 0 ;
    for twol in numtheory[divisors](locn) do
        if type(twol,'even') then
            a := a+numtheory[sigma](locn/twol) ;
        end if;
    end do:
    for fourl in numtheory[divisors](locn) do
        if modp(fourl,4) = 0 then
            a := a-numtheory[sigma](locn/fourl) ;
        end if;
    end do:
    %*2 ;
end proc: # R. J. Mathar, Nov 03 2015

a[n_] := 2*Sum[If[Mod[d,4] == 2, DivisorSigma[1, 2*n/d], 0], {d, Divisors[ 2*n ] } ];
Array[a, 70] (* Jean-François Alcover, Dec 03 2017 *)

A007429(n) = sumdiv(n, d, sigma(d));
a(n) = 2*A007429(n) - if(n%2, 0, 2*A007429(n\2));
vector(70, n, a(n))  \\ Gheorghe Coserea, May 04 2016

A263829 Total number c_{pi_1(B_2)}(n) of n-coverings over the second amphicosm.

Original entry on oeis.org

1, 3, 5, 13, 7, 19, 9, 43, 18, 33, 13, 93, 15, 51, 35, 137, 19, 110, 21, 175, 45, 99, 25, 355, 38, 129, 58, 285, 31, 289, 33, 455, 65, 201, 63, 626, 39, 243, 75, 721, 43, 483, 45, 589, 126, 339, 49, 1305, 66, 498, 95, 783, 55, 750, 91, 1227

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.

Cf. A263825-A263830, A263832.

A001001(n) = sumdiv(n, d, sigma(d) * d);
A007429(n) = sumdiv(n, d, sigma(d));
A007434(n) = sumdiv(n, d, moebius(n\d) * d^2);
A059376(n) = sumdiv(n, d, moebius(n\d) * d^3);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
EpiPcZn(n) = sumdiv(n, d, moebius(n\d) * d^2 * gcd(d,2));
S1(n)      = if (n%2, 0, A001001(n\2));
S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
S22(n)     = { if (n%2, A060640(n), if (n%4, 0,
  sumdiv(n\4, d, 2*d*(sigma(n\(2*d)) - sigma(n\(4*d))))));
};
A027844(n) = S1(n) + S11(n) + S21(n);
a(n) = { 1/n * sumdiv(n, d,
  A059376(d) * S1(n\d) + EpiPcZn(d) * S21(n\d) + A007434(d) * S22(n\d));
};
vector(56, n, a(n))  \\ Gheorghe Coserea, May 04 2016

More terms from Gheorghe Coserea, May 04 2016

A280085 Partial sums of A206032 (Product_{d|n} sigma(d)).

Original entry on oeis.org

1, 4, 8, 29, 35, 179, 187, 502, 554, 878, 890, 29114, 29128, 29704, 30280, 40045, 40063, 113071, 113091, 208347, 209371, 210667, 210691, 25612291, 25612477, 25614241, 25616321, 25842113, 25842143, 52715999, 52716031, 53331226, 53333530, 53336446, 53338750

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

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

Cf. A000203, A007429, A206032, A280077 (partial sums of A007429), A280086 (partial products of A206032).

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

Accumulate@ Array[Product[DivisorSigma[1, d], {d, Divisors@ #}] &, 35] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Sum_{i=1..n} A206032(i).

A280086 Partial products of A206032 (Product_{d|n} sigma(d)).

Original entry on oeis.org

1, 3, 12, 252, 1512, 217728, 1741824, 548674560, 28531077120, 9244068986880, 110928827842560, 3130855237028413440, 43831973318397788160, 25247216631397125980160, 14542396779684744564572160, 142006504553621530673047142400, 2556117081965187552114848563200

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

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

Cf. A000203, A007429, A206032, A280078 (partial products of A007429), A280085 (partial sums of A206032).

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

FoldList[Times[#1, #2] &, Array[Product[DivisorSigma[1, d], {d, Divisors@ #}] &, 17]] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Product_{i=1..n} A206032(i).

A290478 Triangle read by rows in which row n lists the sum of the divisors of each divisor of n.

Original entry on oeis.org

1, 1, 3, 1, 4, 1, 3, 7, 1, 6, 1, 3, 4, 12, 1, 8, 1, 3, 7, 15, 1, 4, 13, 1, 3, 6, 18, 1, 12, 1, 3, 4, 7, 12, 28, 1, 14, 1, 3, 8, 24, 1, 4, 6, 24, 1, 3, 7, 15, 31, 1, 18, 1, 3, 4, 12, 13, 39, 1, 20, 1, 3, 7, 6, 18, 42, 1, 4, 8, 32, 1, 3, 12, 36, 1, 24, 1, 3, 4, 7

Offset: 1

Michel Lagneau, Aug 03 2017

Or, in the triangle A027750(n), replace each element with the sum of its divisors.
The row whose index x is a prime power p^m (p prime and m >= 0) is equal to (1, sigma(p), sigma(p^2), ..., sigma(p^(m-1))).
We observe the following properties of row n when n is the product of k distinct primes, k = 1,2,...:
when n = prime(m), row n = (1, prime(m)+1);
when n is the product of two distinct primes p < q, row n = (1, p+1, q+1,(p+1)(q+1));
when n is the product of three distinct primes p < q < r, row n = (1, p+1, q+1, r+1, (p+1)(q+1), (p+1)(r+1), (q+1)(r+1), sigma(p*q*r));

			Row 6 is (a(11), a(12), a(13), a(14)) = (1, 3, 4, 12) because sigma(A027750(11))= sigma(1) = 1, sigma(A027750(12))= sigma(2) = 3, sigma(A027750(13))= sigma(3) = 4 and sigma(A027750(14)) = sigma(6) = 12.
Triangle begins:
  1;
  1,  3;
  1,  4;
  1,  3,  7;
  1,  6;
  1,  3,  4, 12;
  1,  8;
  1,  3,  7, 15;
  1,  4, 13;
  1,  3,  6, 18;
  ...

Michel Marcus, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened).

Cf. A000203, A027750, A290532.

Cf. A007429 (row sums), A206032 (row products).

[[SumOfDivisors(d): d in Divisors(n)]: n in [1..20]]; // Vincenzo Librandi, Sep 08 2017

with(numtheory):nn:=100:
for n from 1 to nn do:
  d1:=divisors(n):n1:=nops(d1):
   for i from 1 to n1 do:
     s:=sigma(d1[i]):
     printf(`%d, `,s):
   od:
od:

Array[DivisorSigma[1, Divisors@ #] &, 24] // Flatten (* Michael De Vlieger, Aug 07 2017 *)

row(n) = apply(sigma, divisors(n)); \\ Michel Marcus, Dec 27 2021

a(n) = sigma(A027750(n)).

A318845 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} sigma(j), where sigma(j) = sum of divisors of j (A000203).

Original entry on oeis.org

1, 3, 6, 6, 8, 18, 10, 10, 24, 24, 14, 36, 16, 30, 48, 15, 20, 72, 22, 48, 60, 42, 26, 60, 46, 48, 82, 60, 32, 144, 34, 21, 84, 60, 80, 144, 40, 66, 96, 80, 44, 180, 46, 84, 192, 78, 50, 90, 76, 138, 120, 96, 56, 246, 112, 100, 132, 96, 62, 288, 64, 102, 240, 28, 128, 252, 70, 120, 156, 240

Offset: 1

Ilya Gutkovskiy, Sep 04 2018

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

Cf. A000203, A007429, A007430, A288417, A318768.

Table[Sum[(-1)^(n/d + 1) Sum[DivisorSigma[1, j], {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, DivisorSigma[1, #] &] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSum[k, DivisorSigma[1, #] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3); f[2, e_] := (e+1)*(e+2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2025 *)

a(n) = {my(f = factor(n), p , e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, (e+1)*(e+2)/2, (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3)));} \\ Amiram Eldar, May 26 2025

G.f.: Sum_{k>=1} A007429(k)*x^k/(1 + x^k).
L.g.f.: log(Product_{k>=1} (1 + x^k)^(A007429(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
From Amiram Eldar, May 26 2025: (Start)
Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (2*p^(e+3) - (e^2+5*e+6)*p^2 + (2*e^2+8*e+6)*p - e^2 - 3*e -2)/(2*(p-1)^3) for an odd prime p.
Dirichlet g.f: zeta(s-1) * zeta(s)^2 * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^6/864 = 1.112718... . (End)

A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

Original entry on oeis.org

1, 3, 5, 7, 7, 15, 9, 15, 18, 21, 13, 35, 15, 27, 35, 31, 19, 54, 21, 49, 45, 39, 25, 75, 38, 45, 58, 63, 31, 105, 33, 63, 65, 57, 63, 126, 39, 63, 75, 105, 43, 135, 45, 91, 126, 75, 49, 155, 66, 114, 95, 105, 55, 174, 91, 135, 105, 93, 61, 245, 63, 99

Offset: 1

Ilya Gutkovskiy, Sep 13 2019

Inverse Moebius transform of A002131.

Dirichlet convolution of A000027 with A001227.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A001227, A002131, A007429, A026741, A060640, A109386, A133700, A288417, A300707.

nmax = 62; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[Total[Select[Divisors[d], OddQ[d/#] &]], {d, Divisors[n]}]; Table[a[n], {n, 1, 62}]

a(n)={sumdiv(n, d, if(n/d%2, sigma(d)))} \\ Andrew Howroyd, Sep 13 2019

G.f.: Sum_{k>=1} A002131(k) * x^k / (1 - x^k).
G.f.: Sum_{k>=1} A001227(k) * x^k / (1 - x^k)^2.
a(n) = Sum_{d|n} A002131(d).
a(n) = Sum_{d|n} d * A001227(n/d).
a(n) = (A007429(n) + A288417(n)) / 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). - Amiram Eldar, Oct 23 2022

A343442 If n = Product (p_j^k_j) then a(n) = Product (p_j + 2), with a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 15 2021

Cf. A000203, A001615, A007429, A007947, A008683, A048250, A062953, A063441, A074816, A092261, A107758, A107759, A319132, A335005.

a[1] = 1; a[n_] := Times @@ ((#[[1]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 70}]
nmax = 70; CoefficientList[Series[Sum[MoebiusMu[k]^2 DivisorSigma[1, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, moebius(d)^2 * sigma(d)) \\ Andrew Howroyd, Apr 15 2021

G.f.: Sum_{k>=1} mu(k)^2 * sigma(k) * x^k / (1 - x^k), where mu = A008683 and sigma = A000203.
a(n) = Sum_{d|n} mu(d)^2 * sigma(d).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/(12*zeta(3)) = 0.684216... (A335005). - Amiram Eldar, Nov 13 2022
a(n) = Sum_{d|n} mu(d)^2*psi(d), where psi is A001615. - Ridouane Oudra, Jul 24 2025

A344061 a(n) = Sum_{d|n} sigma(d)^(n/d).

Original entry on oeis.org

1, 4, 5, 17, 7, 56, 9, 146, 78, 298, 13, 1501, 15, 2276, 1265, 9219, 19, 25716, 21, 77519, 16929, 177328, 25, 739582, 7808, 1594562, 264382, 5611241, 31, 15699452, 33, 48863172, 4196081, 129140542, 312753, 447589422, 39, 1162261928, 67111665, 3771805472, 43, 10764897556, 45

Offset: 1

Seiichi Manyama, May 08 2021

nonn

Cf. A007429, A055225, A279789, A309369, A342471, A344060.

a[n_] := DivisorSum[n, DivisorSigma[1 , #]^(n/#) &]; Array[a, 43] (* Amiram Eldar, May 08 2021 *)

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

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

G.f.: Sum_{k >= 1} sigma(k) * x^k/(1 - sigma(k) * x^k).

If p is prime, a(p) = 2 + p.

A361706 Inverse Moebius transform applied twice to primes.

Original entry on oeis.org

2, 7, 9, 19, 15, 37, 21, 50, 39, 65, 35, 116, 45, 91, 87, 134, 63, 174, 71, 200, 125, 155, 87, 322, 125, 197, 172, 282, 113, 383, 131, 349, 217, 271, 213, 555, 161, 311, 267, 546, 183, 555, 195, 482, 402, 379, 215, 857, 267, 546, 369, 602, 245, 768, 349, 774, 421, 503, 281, 1204, 287, 561, 582, 875, 425

Offset: 1

Ilya Gutkovskiy, Mar 21 2023

nonn

Dirichlet convolution of primes with the number of divisors function.

Winston de Greef, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A000005, A000040, A007429, A007445, A361707.

a:= (proc(p) proc(n) uses numtheory;
       add(p(d), d=divisors(n))
     end end@@2)(ithprime):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 23 2023

Table[Sum[DivisorSigma[0, n/d] Prime[d], {d, Divisors[n]}], {n, 1, 65}]

a(n) = sumdiv(n, d, numdiv(n/d)*prime(d)); \\ Michel Marcus, Mar 23 2023

G.f.: Sum_{i>=1} Sum_{j>=1} prime(i) * x^(i*j) / (1 - x^(i*j)).

a(n) = Sum_{d|n} A000005(n/d) * prime(d).

cp's OEIS Frontend

A263827 The number c_{Cc pi_1(B_1)}(2n) of the second amphicosm 2n-coverings over the first amphicosm.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A263829 Total number c_{pi_1(B_2)}(n) of n-coverings over the second amphicosm.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A280085 Partial sums of A206032 (Product_{d|n} sigma(d)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A280086 Partial products of A206032 (Product_{d|n} sigma(d)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A290478 Triangle read by rows in which row n lists the sum of the divisors of each divisor of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A318845 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} sigma(j), where sigma(j) = sum of divisors of j (A000203).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A327096 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 - x^(2*k)), where sigma = A000203.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A343442 If n = Product (p_j^k_j) then a(n) = Product (p_j + 2), with a(1) = 1.

Views

Author

Keywords

Links