A048272 -id:A048272 - cp's OEIS frontend

A335021 a(n) = Sum_{d|n, 1 < d < n} (-1)^(d + 1).

0, 0, 0, -1, 0, 0, 0, -2, 1, 0, 0, -2, 0, 0, 2, -3, 0, 0, 0, -2, 2, 0, 0, -4, 1, 0, 2, -2, 0, 0, 0, -4, 2, 0, 2, -3, 0, 0, 2, -4, 0, 0, 0, -2, 4, 0, 0, -6, 1, 0, 2, -2, 0, 0, 2, -4, 2, 0, 0, -4, 0, 0, 4, -5, 2, 0, 0, -2, 2, 0, 0, -6, 0, 0, 4, -2, 2, 0, 0, -6, 3, 0, 0, -4, 2, 0, 2, -4

Offset: 1

Ilya Gutkovskiy, May 19 2020

Number of odd nontrivial divisors of n minus number of even nontrivial divisors of n.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A048272, A070824, A325937, A325939, A335022.

Cf. A001227, A007814.

Table[DivisorSum[n, (-1)^(# + 1) &, 1 < # < n &], {n, 1, 88}]
nmax = 88; CoefficientList[Series[Sum[(-1)^(k + 1) x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d>1) && (dMichel Marcus, May 20 2020

from sympy import divisor_count
def A335021(n): return 0 if n == 1 else (1-(m:=(~n & n-1).bit_length()))*divisor_count(n>>m)-((n&1)<<1) # Chai Wah Wu, Jul 01 2022

G.f.: Sum_{k>=2} (-1)^(k + 1) * x^(2*k) / (1 - x^k).
G.f.: - Sum_{k >= 2} x^(2*k)/(1 + x^k). - Peter Bala, Jan 12 2021
a(n) = A001227(n)*(1 - A007814(n)) - 1 + (-1)^n, if n > 1. - Sebastian Karlsson, Jan 14 2021

A338813 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} (1+x^j)^(u/j).

Original entry on oeis.org

1, 0, 1, 4, 0, 1, -6, 16, 0, 1, 48, -30, 40, 0, 1, 0, 448, -90, 80, 0, 1, 1440, -840, 2128, -210, 140, 0, 1, -10080, 23532, -6720, 7168, -420, 224, 0, 1, 120960, -127008, 177868, -30240, 19488, -756, 336, 0, 1, 0, 2191104, -1018080, 892540, -100800, 45696, -1260, 480, 0, 1

Offset: 1

Seiichi Manyama, Nov 10 2020

Also the Bell transform of A338814.

			exp(Sum_{n>0} u*A048272(n)*x^n/n) = 1 + u*x + u^2*x^2/2! + (4*u+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       0,     1;
       4,     0,     1;
      -6,    16,     0,    1;
      48,   -30,    40,    0,    1;
       0,   448,   -90,   80,    0,   1;
    1440,  -840,  2128, -210,  140,   0, 1;
  -10080, 23532, -6720, 7168, -420, 224, 0, 1;
  ...

Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.

Column k=1 gives A338814.
Row sums give A168243.
Cf. A048272, A075525, A338805.

a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSum[n, (-1)^(# + 1) &]]; T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], Sum[a[j] * Binomial[n - 1, j - 1] * T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, (1+x^j+x*O(x^n))^(u/j)), n), k)}

a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1)))

E.g.f.: exp(Sum_{n>0} u*A048272(n)*x^n/n).
T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} A048272(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} A048272(i_j)/i_j.

A345750 E.g.f.: Product_{k>=1} (1 + (exp(x) - 1)^k)^(1/k).

Original entry on oeis.org

1, 1, 2, 9, 49, 310, 2521, 25557, 290550, 3555041, 48104901, 741103946, 12825399313, 240202011881, 4747281446090, 98808864563065, 2194031697420057, 52582450760730398, 1357237338948268649

Offset: 0

Seiichi Manyama, Jun 26 2021

nonn

Stirling transform of A168243.

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A048272, A048993, A168243, A305550, A305987, A345749, A345751.

max = 18; Range[0, max]! * CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k)^(1/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+(exp(x)-1)^k)^(1/k))))

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, -sumdiv(k, d, (-1)^d)*(exp(x)-1)^k/k))))

E.g.f.: exp( Sum_{k>=1} A048272(k) * (exp(x) - 1)^k / k ).

a(n) = Sum_{k=0..n} Stirling2(n,k) * A168243(k).

A351619 a(n) = Sum_{p|n, p prime} (-1)^p.

Original entry on oeis.org

0, 1, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, -1, 0, -2, 1, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, -1, -1, -1, 1, -2, 0, -2, 0, -1, 0, -2, 0, -1, -1, -1, 0, -2, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -2, 0, -1, -1, -1, 0, -2, 1, -2, -1, -1, 0, -2, -1, -1, 0, -1, 0, -2, 0, -2, -1, -1, 0, -1, 0, -1, -1, -2, 0, -2, 0, -1, -1, -2, 0, -2, 0, -2, 0, -1, 0, -2, 0, -1

Offset: 1

Seiichi Manyama, Mar 02 2022

sign

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

Cf. A001221, A048272, A305614.

A351619[n_] := 2*Boole[EvenQ[n]] - PrimeNu[n]; Array[A351619, 100] (* Paolo Xausa, Jan 28 2025 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, (-1)^f[k, 1]);

my(N=99, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, isprime(k)*(-x)^k/(1-x^k))))

from sympy import primefactors
def A351619(n): return (0 if n%2 else 2) - len(primefactors(n)) # Chai Wah Wu, Mar 02 2022

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

a(n) = -A001221(n) if n is odd and a(n) = 2 - A001221(n) if n is even. - Chai Wah Wu, Mar 02 2022

A364343 Expansion of Sum_{k>0} k * x^k/(1 + x^k)^3.

Original entry on oeis.org

1, -1, 9, -12, 20, -12, 35, -60, 72, -30, 77, -132, 104, -56, 210, -256, 170, -117, 209, -320, 378, -132, 299, -672, 425, -182, 594, -588, 464, -360, 527, -1040, 858, -306, 910, -1224, 740, -380, 1170, -1640, 902, -672, 989, -1364, 1890, -552, 1175, -2928, 1470, -775, 1938, -1872, 1484, -1080, 2090

Offset: 1

Seiichi Manyama, Jul 19 2023

sign

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

Cf. A320900, A364351.

Cf. A002129, A048272, A309731.

a[n_] := DivisorSum[n, (-1)^(# + 1)*(# + 1) &] * n/2; Array[a, 55] (* Amiram Eldar, Jul 20 2023 *)

my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1+x^k)^3))

a(n) = (n/2) * Sum_{d|n} (-1)^(d+1) * (d+1) = (n/2) * (A002129(n) + A048272(n)).

A364351 Expansion of Sum_{k>0} k^2 * x^k/(1 + x^k)^3.

Original entry on oeis.org

1, 1, 15, -6, 40, 12, 77, -60, 180, 30, 187, -120, 260, 56, 630, -376, 442, 117, 551, -340, 1218, 132, 805, -1104, 1325, 182, 1998, -672, 1276, 360, 1457, -2032, 2970, 306, 3290, -1710, 2072, 380, 4134, -3080, 2542, 672, 2795, -1672, 7830, 552, 3337, -6816, 4998, 775, 7038, -2340, 4240, 1080

Offset: 1

Seiichi Manyama, Jul 19 2023

sign

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

Cf. A320900, A364343.

Cf. A000593, A048272, A309732.

a[n_] := DivisorSum[n, (-1)^(n/#+1) * (#+n) &] * n/2; Array[a, 55] (* Amiram Eldar, Jul 20 2023 *)

my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, k^2*x^k/(1+x^k)^3))

a(n) = (n/2) * Sum_{d|n} (-1)^(n/d+1) * (d+n) = (n/2) * (A000593(n) + n * A048272(n)).

A373335 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2k) + x^(3k) + x^(4*k)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 2, 0, 1, -1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, -1, 1, 1, 2, 0, 2, -1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 2, 0, -1, 1, -1, 2, 0, 1, -1, 1, 1, 0, 2, 1, 1, 1, 0, 1, -1, 0, 1, 0, 0, 1, 1, 1, 0, 2, -1, 1, 1, 0, -1, 2, 0, 2

Offset: 1

Seiichi Manyama, Jun 01 2024

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

Cf. A002324, A048272, A373336.

Cf. A001876, A001877.

my(N=110, x='x+O('x^N)); Vec(sum(k=1, N, x^k*(1-x^k)/(1-x^(5*k))))

PARI
```
a(n) = sumdiv(n, d, (d%5==1)-(d%5==2));
```

G.f.: Sum_{k>=1} x^k * (1 - x^k) / (1 - x^(5*k)).

a(n) = A001876(n) - A001877(n).

A373336 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2k) + x^(3k) + x^(4k) + x^(5k) + x^(6*k)).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, -1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 2, 0, 1, -1, 1, 0, 1, 1, 0, 0, 1, -1, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 2, 0, 1, -1, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 1, -1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, -1, 1, 1, 2

Offset: 1

Seiichi Manyama, Jun 01 2024

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

Cf. A002324, A048272, A373335.

Cf. A279061, A363795.

my(N=110, x='x+O('x^N)); Vec(sum(k=1, N, x^k*(1-x^k)/(1-x^(7*k))))

PARI
```
a(n) = sumdiv(n, d, (d%7==1)-(d%7==2));
```

G.f.: Sum_{k>=1} x^k * (1 - x^k) / (1 - x^(7*k)).

a(n) = A279061(n) - A363795(n).

A253951 A partial double sum of integers: a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T = A051731A127093Transpose(A054524) and T(n,1)=0 (* stands for matrix multiplication).

Original entry on oeis.org

0, 1, 5, 9, 20, 23, 42, 52, 69, 77, 113, 119, 165, 177, 190, 214, 279, 291, 366, 379, 399, 422, 517, 533, 599, 625, 679, 701, 829, 846, 986, 1035, 1069, 1105, 1137, 1164, 1339, 1380, 1417, 1449, 1646, 1674, 1883, 1918, 1955, 2008, 2239, 2274, 2420, 2462, 2515, 2559, 2827, 2874, 2929

Offset: 1

Mats Granvik, Jan 20 2015

nonn

a(n) ~ log(A003418(n))*n, based on the comment by Hans Havermann in A048272 referring to an argument by Gareth McCaughan.
The exact relation is: lim_{n->Infinity} log(A003418(k))*n = Sum_{x=1..n} Sum_{y=1..k} T(x,y), where T is the matrix product: T = A051731*A127093*Transpose(A054524) and T(n,1)=0.
Compare a(n) to round(log(A003418)*n)= 0, 1, 5, 10, 20, 25, 42, 54, 70, 78,...

Robert G. Wilson v, Table of n, a(n) for n = 1..1002

with(LinearAlgebra):
N:= 200:
A051731:= Matrix(N,N,(n,k) -> `if`(n mod k = 0, 1, 0),shape=triangular[lower]):
A127093:= Matrix(N,N,(n,k) -> `if`(n mod k = 0, k, 0), shape=triangular[lower]):
A054524T:= Matrix(N,N,(k,n) -> `if`(n mod k = 0, numtheory:-mobius(k),0), shape=triangular[upper]):
T:= A051731 . A127093 . A054524T:
a[1]:= 0:
for n from 2 to N do
  a[n]:= a[n-1] + add(T[i,n],i=1..n) + add(T[n,j],j=2..n-1)
od:
seq(a[n],n=1..N); # Robert Israel, Jan 20 2015

nn = 55;
Z = Table[ If[ Mod[n, k] == 0, 1, 0], {n, nn}, {k, nn}];
A = Table[ If[ Mod[n, k] == 0, k, 0], {n, nn}, {k, nn}];
B = Table[ If[ Mod[n, k] == 0, MoebiusMu[k], 0], {n, nn}, {k, nn}];
MatrixForm[T = Z.A.Transpose[B]];
T[[All, 1]] = 0;
a = Table[ Total[ T[[1 ;; n, 1 ;; n]], 2], {n, nn}]
(* shows a graph *) Show[ ListLinePlot[a], ListLinePlot[ Accumulate[ MangoldtLambda[ Range[ nn]]]]]

a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T=A051731*A127093*Transpose(A054524) and T(n,1)=0. (* stands for matrix multiplication)

A274352 Convolution of A015723 and A000700.

Original entry on oeis.org

0, 1, 2, 4, 7, 10, 18, 26, 36, 53, 76, 104, 140, 190, 252, 336, 437, 564, 732, 936, 1186, 1504, 1894, 2366, 2950, 3659, 4520, 5564, 6822, 8330, 10152, 12326, 14906, 17996, 21662, 25996, 31135, 37190, 44314, 52704, 62532, 74036, 87504, 103212, 121496, 142798

Offset: 0

R. J. Mathar, Jun 18 2016

nonn

Also the convolution of A080054 and A048272.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, corollary 3.4

Cf. A015723, A000700, A080054, A048272.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
     [0, 2, -1, 2][1+irem(d, 4)], d=divisors(j))*b(n-j), j=1..n)/n)
    end:
g:= proc(n) option remember; add((-1)^(d+1), d=divisors(n)) end:
a:= n-> add(b(j)*g(n-j), j=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 18 2016

q[n_, k_] := q[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, q[n - k, k] + q[n - k, k - 1]]]; Table[Sum[SeriesCoefficient[Product[1 + x^j, {j, 1, k, 2}], {x, 0, k}] Sum[i q[#, i], {i, 1, Floor[(Sqrt[8 # + 1] - 1)/2]}] &[n - k], {k, 0, n}], {n, 0, 45}] (* Michael De Vlieger, Jun 18 2016, after Vaclav Kotesovec at A015723 and Vladimir Reshetnikov at A000700 *)

a(n) = Sum_{k=0..n} A015723(k)*A000700(n-k).

a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(3/4) * n^(1/4)). - Vaclav Kotesovec, Sep 14 2021

cp's OEIS Frontend

A335021 a(n) = Sum_{d|n, 1 < d < n} (-1)^(d + 1).

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A338813 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} (1+x^j)^(u/j).

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A345750 E.g.f.: Product_{k>=1} (1 + (exp(x) - 1)^k)^(1/k).

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A351619 a(n) = Sum_{p|n, p prime} (-1)^p.

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A364343 Expansion of Sum_{k>0} k * x^k/(1 + x^k)^3.

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A364351 Expansion of Sum_{k>0} k^2 * x^k/(1 + x^k)^3.

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A373335 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2*k) + x^(3*k) + x^(4*k)).

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A338813 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} (1+x^j)^(u/j).

A373335 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2k) + x^(3k) + x^(4*k)).

A373336 Expansion of Sum_{k>=1} x^k / (1 + x^k + x^(2k) + x^(3k) + x^(4k) + x^(5k) + x^(6*k)).

A253951 A partial double sum of integers: a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T = A051731A127093Transpose(A054524) and T(n,1)=0 (* stands for matrix multiplication).