A300415 -id:A300415 - cp's OEIS frontend

A352047 Sum of the divisor complements of the odd proper divisors of n.

0, 2, 3, 4, 5, 8, 7, 8, 12, 12, 11, 16, 13, 16, 23, 16, 17, 26, 19, 24, 31, 24, 23, 32, 30, 28, 39, 32, 29, 48, 31, 32, 47, 36, 47, 52, 37, 40, 55, 48, 41, 64, 43, 48, 77, 48, 47, 64, 56, 62, 71, 56, 53, 80, 71, 64, 79, 60, 59, 96, 61, 64, 103, 64, 83, 96, 67, 72, 95, 96, 71, 104

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 12; a(10) = 10 * Sum_{d|10, d<10, d odd} 1 / d = 10 * (1/1 + 1/5) = 12.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000035, A000593, A002131, A006519, A111003, A300415.

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), this sequence (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

A352047 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if type(d,'odd') and d < n then
            a := a+n/d ;
        end if;
    end do:
    a ;
end proc:
seq(A352047(n),n=1..30) ; # R. J. Mathar, Mar 09 2022

Table[n DivisorSum[n, 1/# &, # < n && OddQ[#] &], {n, 72}] (* Michael De Vlieger, Mar 02 2022 *)
a[n_] := DivisorSigma[-1, n / 2^IntegerExponent[n, 2]] * n - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n*sumdiv(n, d, if ((d%2) && (dMichel Marcus, Mar 02 2022

a(n) = n * sigma(n >> valuation(n, 2), -1) - n % 2; \\ Amiram Eldar, Oct 13 2023

from math import prod
from sympy import factorint
def A352047(n): return prod(p**e if p == 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) - n % 2 # Chai Wah Wu, Mar 02 2022

a(n) = n * Sum_{d|n, d

a(2n+1) = A000593(2n+1) - 1. - Chai Wah Wu, Mar 01 2022

G.f.: Sum_{k>=2} k * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

exp( 2*Sum_{n>=1} a(n)*x^n/n ) is the g.f. of A300415. - Paul D. Hanna, May 15 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A000593(n) * A006519(n) - A000035(n).

Sum_{k=1..n} a(k) = c * n^2 / 2, where c = Pi^2/8 = 1.23370055... (A111003). (End)

A300413 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 - x^prime(k)).

Original entry on oeis.org

1, 0, 2, 2, 2, 6, 4, 10, 10, 14, 20, 22, 32, 38, 48, 60, 74, 90, 112, 134, 164, 196, 236, 282, 336, 398, 472, 554, 652, 766, 890, 1046, 1206, 1408, 1624, 1876, 2168, 2486, 2860, 3276, 3744, 4282, 4878, 5554, 6316, 7160, 8124, 9186, 10388, 11722, 13216, 14876, 16732, 18794, 21084, 23636

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Convolution of the sequences A000586 and A000607.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000040, A000586, A000607, A015128, A080054, A103265, A280263, A280366, A300414, A300415.

nmax = 55; CoefficientList[Series[Product[(1 + x^Prime[k])/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000040(k))/(1 - x^A000040(k)).

log(a(n)) ~ Pi*sqrt(2*n/log(n/3)) * (1/3 + 2*sqrt(log(n/3) / log(2*n/3)) / 3). - Vaclav Kotesovec, Jan 12 2021

A300414 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))/(1 - x^Fibonacci(k)).

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 30, 42, 62, 84, 114, 154, 198, 260, 332, 418, 530, 654, 810, 994, 1202, 1462, 1752, 2094, 2500, 2948, 3486, 4092, 4776, 5582, 6468, 7490, 8650, 9928, 11406, 13036, 14862, 16934, 19196, 21758, 24592, 27706, 31216, 35038, 39284, 43990, 49100, 54798, 61008, 67798

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Convolution of the sequences A000119 and A003107.

Index entries for related partition-counting sequences

Cf. A000045, A000119, A003107, A015128, A080054, A103265, A280263, A280366, A300413, A300415.

nmax = 49; CoefficientList[Series[Product[(1 + x^Fibonacci[k])/(1 - x^Fibonacci[k]), {k, 2, 20}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 + x^A000045(k))/(1 - x^A000045(k)).

A319672 a(n) = [x^n] Product_{k>=2} ((1 + x^k)/(1 - x^k))^n.

Original entry on oeis.org

1, 0, 4, 6, 40, 110, 520, 1778, 7568, 28320, 116224, 453046, 1837600, 7306234, 29565848, 118786526, 481192480, 1945153838, 7895908852, 32046260282, 130370798320, 530650047710, 2163191769336, 8824509524082, 36037768384832, 147277910160160, 602398740105712, 2465582764631334

Offset: 0

Ilya Gutkovskiy, Sep 25 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000203, A270919, A300415, A319670, A319671.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^n , {k, 2, n}], {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[((1 - x)/((1 + x) EllipticTheta[4, 0, x]))^n, {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k] + (-1)^k - 1) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 27}]

a(n) = [x^n] ((1 - x)/((1 + x)*theta_4(x)))^n, where theta_4() is the Jacobi theta function.
a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - sigma(k) + (-1)^k - 1)*x^k/k).
a(n) ~ c * d^n / sqrt(n), where d = 4.16958962845360844086951404338054148667024... and c = 0.23380422010834870751549442953816486722... - Vaclav Kotesovec, Oct 06 2018

A308654 The overpartition triangle: T(n,k) is the number of overpartitions of n with exactly k positive integer parts, 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 2, 4, 2, 0, 2, 6, 4, 2, 0, 2, 8, 8, 4, 2, 0, 2, 10, 14, 8, 4, 2, 0, 2, 12, 20, 16, 8, 4, 2, 0, 2, 14, 28, 26, 16, 8, 4, 2, 0, 2, 16, 38, 40, 28, 16, 8, 4, 2, 0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2, 0, 2, 20, 60, 84, 72, 48, 28, 16, 8, 4, 2

Offset: 0

Gregory L. Simay, Jun 14 2019

T(n,0) = A000007(n).
T(n,1) = A040000(n) for n > 0.
T(n,2) = A005843(n-1).
T(n,3) = 2*A007980(n-3).
T(n,4) = 2*A061866(n-1).
T(n,5) = 2*A091773(n-5).
Conjecture: T(n,k) = 2*(the associated Poincaré series). If T(n,1) were 1 for n>0, then T(n, k>1) would be a Poincaré series.

			T(5,3) = 8 and counts the overpartitions 3,1,1; 3',1,1; 3,1',1; 3',1',1; 2,2,1; 2',2,1; 2,2,1' and 2',2,1'.
T(16,5) = 404 = T(11,5) + 2*( T(11,4) + T(11,3) + T(11,2) + T(11,1)) = 72 + 2*(84 + 60 + 20 + 2) = 404.
T(16, 5) = T(15,4) + T(11,4) + T(10,4) + T(6,4) + T(5,4) = 248 + 84 + 60 + 8 + 4 = 404.
T(9,1) + T(8,2) + T(7,3) + T(6,4) + T(6,5)= 2 + 14 + 20 + 8 + 2 = 46 =A300415(10).
Triangle: T(n,k) begins:
  1;
  0, 2;
  0, 2,  2;
  0, 2,  4,  2;
  0, 2,  6,  4,  2;
  0, 2,  8,  8,  4,  2;
  0, 2, 10, 14,  8,  4,  2;
  0, 2, 12, 20, 16,  8,  4,  2;
  0, 2, 14, 28, 26, 16,  8,  4, 2;
  0, 2, 16, 38, 40, 28, 16,  8, 4, 2;
  0, 2, 18, 48, 60, 46, 28, 16, 8, 4, 2;
  ...

Row sums give A015128.
Main diagonal T(n,n) gives A040000.
Cf. A005843, A007980, A061866, A091773, A235792, A300415.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      expand(`if`(j>0, 2*x^j, 1)*b(n-i*j, i-1)), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Jun 15 2019

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Expand[If[j > 0, 2*x^j, 1]*b[n - i*j, i - 1]], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P. Heinz *)

Sum_{k=0..n} T(n,k) = A015128(n), the number of overpartitions of n.
If k > n, T(n,k) = 0.
If n >= k > n/2, T(n,k) = 2*A015128(n-k).
Conjecture: T(n,k) = T(n-k, k) + 2*(T(n-k, k-1) + ... + T(n-k, 1)).
Conjecture: T(n,k) = T(n-1, k-1) + T(n-k, k-1) + T(n-k-1, k-1) + T(n-2k, k-1) + T(n-2k-1) + ...
Conjecture: T(n,1) + T(n-1,2) + ... + T(n-floor(n/2),floor(n/2)) = A300415(n+1).
T(n,2) = 2n - 2.
Conjecture: g.f. T(n,k) = 2*(1+x)(1+x^2)...(1+x^(k-1))/((1-x)...(1-x^k)).
Sum_{k=1..n} k * T(n,k) = A235792(n). - Alois P. Heinz, Jun 15 2019

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cp's OEIS Frontend

A352047 Sum of the divisor complements of the odd proper divisors of n.

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A300413 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 - x^prime(k)).

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A300414 Expansion of Product_{k>=2} (1 + x^Fibonacci(k))/(1 - x^Fibonacci(k)).

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A319672 a(n) = [x^n] Product_{k>=2} ((1 + x^k)/(1 - x^k))^n.

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A308654 The overpartition triangle: T(n,k) is the number of overpartitions of n with exactly k positive integer parts, 0 <= k <= n.

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