A129527 -id:A129527 - cp's OEIS frontend

A129266 Duplicate of A129527.

1, 3, 3, 7, 5, 9, 7, 15, 9, 15

Offset: 1

dead

A115361 Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).

1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Jan 21 2006

Row sums are the 'ruler function' A001511. Columns are stretched Fredholm-Rueppel sequences (A036987). Inverse is A115359.
Eigensequence of triangle A115361 = A018819 starting with offset 1: (1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, ...). - Gary W. Adamson, Nov 21 2009
From Gary W. Adamson, Nov 27 2009: (Start)
A115361 * [1, 2, 3, ...] = A129527 = (1, 3, 3, 7, 5, 9, 7, 15, ...).
(A115361)^(-1) * [1, 2, 3, ...] = A115359 * [1, 2, 3, ...] = A026741 starting /Q (1, 1, 3, 2, 5, 3, 7, 4, 9, ...). (End)
This is the lower-left triangular part of the inverse of the infinite matrix A_{ij} = [i=j] - [i=2j], its upper-right part (above / right to the diagonal) being zero. The n-th row has 1 in column n/2^i, i = 0, 1, ... as long as this is an integer. - M. F. Hasler, May 13 2018
The rows are the reversed binary expansions of A127804. - Tilman Piesk, Jun 10 2025

			Triangle begins:
  1;
  1,1;
  0,0,1;
  1,1,0,1;
  0,0,0,0,1;
  0,0,1,0,0,1;
  0,0,0,0,0,0,1;
  1,1,0,1,0,0,0,1;
  0,0,0,0,0,0,0,0,1;
  0,0,0,0,1,0,0,0,0,1;
  0,0,0,0,0,0,0,0,0,0,1;

Tilman Piesk, Illustration of first 32 rows.

Cf. A016741, A018819, A129527. - Gary W. Adamson, Nov 21 2009

Cf. A036987, A209229, A127804.

A115361 := proc(n,k)
    for j from 0 do
        if k+(2*j-1)*(k+1) > n then
            return 0 ;
        elif k+(2^j-1)*(k+1) = n then
            return 1 ;
        end if;
    end do;
end proc: # R. J. Mathar, Jul 14 2012

(*recurrence*)
Clear[t]
t[1, 1] = 1;
t[n_, k_] :=
t[n, k] =
  If[k == 1, Sum[t[n, k + i], {i, 1, 2 - 1}],
   If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 14}]] (* Mats Granvik, Jun 26 2014 *)

tabl(nn) = {T = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = T^(-1); for (n=1, nn, for (k=1, n, print1(Ti[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

A115361_row(n,v=vector(n))={until(bittest(n,0)||!n\=2,v[n]=1);v} \\ Yields the n-th row (of length n). - M. F. Hasler, May 13 2018

T(n,k)={if(n%k, 0, my(e=valuation(n/k,2)); n/k==1<Andrew Howroyd, Aug 03 2018

# translation of Maple code by R. J. Mathar
def a115361(n, k):
    j = 0
    while True:
        if k + (2*j - 1) * (k + 1) > n:
            return 0
        elif k + (2**j - 1) * (k + 1) == n:
            return 1
        else:
            j += 1  #  Tilman Piesk, Jun 10 2025

Number triangle whose k-th column has g.f. x^k*sum{j>=0} x^((2^j-1)*(k+1)).

T(n,k) = A209229((n+1)/(k+1)) for k+1 divides n+1, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 05 2018

A327625 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^2.

Original entry on oeis.org

1, 2, 4, 4, 5, 8, 7, 8, 13, 10, 11, 16, 13, 14, 20, 16, 17, 26, 19, 20, 28, 22, 23, 32, 25, 26, 40, 28, 29, 40, 31, 32, 44, 34, 35, 52, 37, 38, 52, 40, 41, 56, 43, 44, 65, 46, 47, 64, 49, 50, 68, 52, 53, 80, 55, 56, 76, 58, 59, 80, 61, 62, 91, 64, 65, 88, 67, 68, 92, 70

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

Sum of divisors d of n such that n/d is power of 3.

Inverse Moebius transform of A195459.

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

Cf. A000010, A000244, A001651 (fixed points), A051064, A129527, A195459.

[(1/2)*&+[EulerPhi(3*d) :d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Sep 19 2019

nmax = 70; CoefficientList[Series[Sum[x^(3^k)/(1 - x^(3^k))^2, {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[Log[3, n/#]] &]; Table[a[n], {n, 1, 70}]
a[n_] := 1/2 Sum[EulerPhi[3 d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

A327625(n) = (n+sumdiv(n,d,my(b=0); if(isprimepower(n/d,&b)&&(3==b),d,0))); \\ Antti Karttunen, Sep 19 2019

G.f. A(x) satisfies: A(x) = A(x^3) + x/(1 - x)^2.
G.f.: Sum_{k>=1} phi(3*k) * x^k / (2 * (1 - x^k)), where phi = A000010.
a(n) = (1/2) * Sum_{d|n} phi(3*d).
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(3^e) = (3^(e+1)-1)/2, and a(p^e) = p^e for p != 3.
Sum_{k=1..n} a(k) ~ (9/16) * n^2. (End)
Dirichlet g.f.: zeta(s-1)*(1+1/(3^s-1)). - Amiram Eldar, Dec 17 2022

A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 7, 8, 9, 12, 11, 12, 13, 14, 18, 16, 17, 18, 19, 24, 21, 22, 23, 24, 31, 26, 27, 28, 29, 36, 31, 32, 33, 34, 42, 36, 37, 38, 39, 48, 41, 42, 43, 44, 54, 46, 47, 48, 49, 62, 51, 52, 53, 54, 66, 56, 57, 58, 59, 72, 61, 62, 63, 64, 78, 66, 67, 68, 69, 84, 71, 72, 73, 74, 93, 76

Offset: 1

Seiichi Manyama, Dec 16 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A129527, A327625, A359099, A373188.

Cf. A000010, A055457.

Array[DivisorSum[#, EulerPhi[5 #] &]/4 &, 76] (* Michael De Vlieger, Dec 16 2022 *)
f[p_, e_] := If[p == 5, (5^(e + 1) - 1)/4, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)

PARI
```
a(n) = sumdiv(n, d, eulerphi(5*d))/4;
```

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(5*k)*x^k/(1-x^k))/4)

G.f.: Sum_{k>=1} phi(5 * k) * x^k / (4 * (1 - x^k)).
G.f.: Sum_{k>=0} x^(5^k) / (1 - x^(5^k))^2.
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = p if p != 5.
Dirichlet g.f.: zeta(s-1)*(1+1/(5^s-1)).
Sum_{k=1..n} a(k) ~ (25/48) * n^2. (End)
From Seiichi Manyama, Jun 04 2024: (Start)
G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^5).
If n == 0 (mod 5), a(n) = n + a(n/5) otherwise a(n) = n. (End)

A359099 a(n) = (1/6) * Sum_{d|n} phi(7 * d).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 8, 9, 10, 11, 12, 13, 16, 15, 16, 17, 18, 19, 20, 24, 22, 23, 24, 25, 26, 27, 32, 29, 30, 31, 32, 33, 34, 40, 36, 37, 38, 39, 40, 41, 48, 43, 44, 45, 46, 47, 48, 57, 50, 51, 52, 53, 54, 55, 64, 57, 58, 59, 60, 61, 62, 72, 64, 65, 66, 67, 68, 69, 80, 71, 72, 73, 74, 75, 76, 88, 78, 79

Offset: 1

Seiichi Manyama, Dec 16 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A129527, A327625, A359100, A373188.

Cf. A000010, A373217.

Array[DivisorSum[#, EulerPhi[7 #] &]/6 &, 79] (* Michael De Vlieger, Dec 16 2022 *)
f[p_, e_] := If[p == 7, (7^(e + 1) - 1)/6, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)

PARI
```
a(n) = sumdiv(n, d, eulerphi(7*d))/6;
```

my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(7*k)*x^k/(1-x^k))/6)

G.f.: Sum_{k>=1} phi(7 * k) * x^k / (6 * (1 - x^k)).
G.f.: Sum_{k>=0} x^(7^k) / (1 - x^(7^k))^2.
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(7^e) = (7^(e+1)-1)/6, and a(p^e) = p if p != 7.
Dirichlet g.f.: zeta(s-1)*(1+1/(7^s-1)).
Sum_{k=1..n} a(k) ~ (49/96) * n^2. (End)
From Seiichi Manyama, Jun 04 2024: (Start)
G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^7).
If n == 0 (mod 7), a(n) = n + a(n/7) otherwise a(n) = n. (End)

A385134 The sum of divisors d of n such that n/d is a biquadratefree number (A046100).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 30, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 60, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 120, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 120, 84, 144

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A000203, A008683, A046100, A307430, A385006.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), this sequence (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := p^(e-3)*(1 + p + p^2 + p^3); f[p_, 1] := 1 + p; f[p_, 2] := 1 + p + p^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^max(e-3,0) * (p^min(e+1,4)-1)/(p-1));}

a(n) = Sum_{d | n} d * A307430(n/d) = n * Sum_{d | n} A307430(d) / d.
a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3), where mu is the Moebius function (A008683).
Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = p^(e-3) * (1 + p + p^2 + p^3), for e >= 3.
In general, the sum of divisors d of n such that n/d is k-free (not divisible by a k-th power larger than 1) is multiplicative with a(p^e) =  p^max(e-k+1,0) * (p^min(e+1,k)-1)/(p-1).
Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(4*s).
In general, the sum of divisors d of n such that n/d is k-free has Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(k*s).
Sum_{i=1..n} a(i) ~ (1575 / (2*Pi^6)) * n^2.

A385135 The sum of divisors d of n such that n/d is an exponentially odd number (A268335).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 13, 12, 18, 12, 24, 14, 24, 24, 26, 18, 36, 20, 36, 32, 36, 24, 52, 30, 42, 37, 48, 30, 72, 32, 53, 48, 54, 48, 72, 38, 60, 56, 78, 42, 96, 44, 72, 72, 72, 48, 104, 56, 90, 72, 84, 54, 111, 72, 104, 80, 90, 60, 144, 62, 96, 96, 106, 84, 144

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A013662, A033634, A268335.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), this sequence (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := p^e + (p^(e+1) - If[EvenQ[e], p, 1])/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^e + (p^(e + 1) - if(e%2, 1, p))/(p^2 - 1));}

Multiplicative with a(p^e) = p^e + (p^(e+1) - 1)/(p^2-1) if e is odd, and p^e + (p^(e+1) - p)/(p^2-1) if e is even.
Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.542116283140158741... .

A385136 The sum of divisors d of n such that n/d is a cubefull number (A036966).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 19, 17, 18, 19, 20, 21, 22, 23, 27, 25, 26, 28, 28, 29, 30, 31, 39, 33, 34, 35, 36, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 57, 49, 50, 51, 52, 53, 56, 55, 63, 57, 58, 59, 60, 61, 62, 63, 79, 65, 66, 67, 68

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A013661, A036966, A385005.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), this sequence (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := (p^(e+1) - p^e + p^(e-2) - 1)/(p-1); f[p_, 1] := p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e == 1, p, (p^(e+1) - p^e + p^(e-2) - 1)/(p-1)));}

Multiplicative with a(p) = p and a(p^e) = (p^(e+1) - p^e + p^(e-2) - 1)/(p-1) for e >= 2.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^6) = 1.022486596136980366... .

A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

Original entry on oeis.org

1, 3, 4, 7, 5, 12, 7, 15, 13, 15, 11, 28, 13, 21, 20, 31, 17, 39, 19, 35, 28, 33, 23, 60, 25, 39, 40, 49, 29, 60, 31, 63, 44, 51, 35, 91, 37, 57, 52, 75, 41, 84, 43, 77, 65, 69, 47, 124, 49, 75, 68, 91, 53, 120, 55, 105, 76, 87, 59, 140, 61, 93, 91, 127, 65, 132

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A003586, A064987, A072044, A072045, A072079.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), this sequence (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p < 5, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p < 5, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385138(n).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 3, and p^e if p >= 5.
In general, the sum of divisors d of n such that n/d is q-smooth (not divisible by a prime larger than q) is multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= q, and p^e if p > q.
Dirichlet g.f.: zeta(s-1) / ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-smooth has Dirichlet g.f.: zeta(s-1) / Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (3/4)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-smooth has an average order c * n^2 / 2, where c = A072044(k-1)/A072045(k-1) for k >= 2.

A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 8, 8, 9, 12, 12, 12, 14, 16, 18, 16, 18, 18, 20, 24, 24, 24, 24, 24, 31, 28, 27, 32, 30, 36, 32, 32, 36, 36, 48, 36, 38, 40, 42, 48, 42, 48, 44, 48, 54, 48, 48, 48, 57, 62, 54, 56, 54, 54, 72, 64, 60, 60, 60, 72, 62, 64, 72, 64, 84, 72, 68, 72

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A007310, A013661, A064987, A072044, A072045, A186099.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), this sequence (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p > 3, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p > 3, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385137(n).
Multiplicative with a(p^e) = p^e if p <= 3, and (p^(e+1)-1)/(p-1) if p >= 5.
In general, the sum of divisors d of n such that n/d is q-rough (not divisible by a prime smaller than q) is multiplicative with a(p^e) = p^e if p <= q, and (p^(e+1)-1)/(p-1) if p > q.
Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-rough has Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (Pi^2/18)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-rough has an average order c * n^2 / 2, where c = zeta(2) * A072045(k-1)/A072044(k-1) for k >= 2.

cp's OEIS Frontend

A129266 Duplicate of A129527.

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A115361 Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).

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Maple

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A327625 Expansion of Sum_{k>=0} x^(3^k) / (1 - x^(3^k))^2.

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Magma

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A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).

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A359099 a(n) = (1/6) * Sum_{d|n} phi(7 * d).

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A385134 The sum of divisors d of n such that n/d is a biquadratefree number (A046100).

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Mathematica

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A385135 The sum of divisors d of n such that n/d is an exponentially odd number (A268335).

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A385136 The sum of divisors d of n such that n/d is a cubefull number (A036966).

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