A026741 -id:A026741 - cp's OEIS frontend

A118392 Denominator of sum of reciprocals of first n tetrahedral numbers A000292.

1, 4, 20, 5, 7, 56, 24, 15, 55, 44, 52, 91, 35, 80, 272, 51, 57, 380, 140, 77, 253, 184, 200, 325, 117, 252, 812, 145, 155, 992, 352, 187, 595, 420, 444, 703, 247, 520, 1640, 287, 301, 1892, 660, 345, 1081, 752, 784, 1225, 425, 884, 2756, 477, 495, 3080

Offset: 1

Jonathan Vos Post, Apr 27 2006

Numerators are A118391.

			a(1) = 1 = denominator of 1/1.
a(2) = 4 = denominator of 5/4 = 1/1 + 1/4.
a(3) = 20 = denominator of 27/20 = 1/1 + 1/4 + 1/10.
a(4) = 5 = denominator of 7/5 = 1/1 + 1/4 + 1/10 + 1/20.
a(5) = 7 = denominator of 10/7 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35.
a(20) = 77 = denominator of 115/77 = 1/1 + 1/4 + 1/10 + 1/20 + 1/35 + 1/56 + 1/84 + 1/120 + 1/165 + 1/220 + 1/286 + 1/364 + 1/455 + 1/560 + 1/680 + 1/816 + 1/969 + 1/1140 + 1/1330 + 1/1540.
Fractions are: 1/1, 5/4, 27/20, 7/5, 10/7, 81/56, 35/24, 22/15, 81/55, 65/44, 77/52, 135/91, 52/35, 119/80, 405/272, 76/51, 85/57, 567/380, 209/140, 115/77, 378/253, 275/184, 299/200, 486/325, 175/117, 377/252, 1215/812, 217/145, 232/155, 1485/992.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000292, A022998, A026741, A118391.

[Denominator(3*n*(n+3)/(2*(n+1)*(n+2))): n in [1..60]]; // G. C. Greubel, Feb 18 2021

A118392:= n -> denom(3*n*(n+3)/(2*(n+1)*(n+2)));
seq(A118392(n), n = 1..60); # G. C. Greubel, Feb 18 2021

Accumulate[1/Binomial[Range[70]+2,3]]//Denominator (* Harvey P. Dale, Jun 07 2018 *)

s=0;for(i=3,50,s+=1/binomial(i,3);print(denominator(s))) /* Phil Carmody, Mar 27 2012 */

[denominator(3*n*(n+3)/(2*(n+1)*(n+2))) for n in (1..60)] # G. C. Greubel, Feb 18 2021

A118391(n)/A118392(n) = Sum_{i=1..n} 1/A000292(n).
A118391(n)/A118392(n) = Sum_{i=1..n} 1/C(n+2,3).
A118391(n)/A118392(n) = Sum_{i=1..n} 6/(n*(n+1)*(n+2)).
a(n) = denominator( 3*n*(n+3)/(2*(n+1)*(n+2)) ). - G. C. Greubel, Feb 18 2021

More terms from Harvey P. Dale, Jun 07 2018

A130334 Smallest m>n such that the m-th and n-th triangular numbers are coprime.

Original entry on oeis.org

2, 4, 10, 6, 7, 10, 9, 10, 13, 12, 13, 22, 15, 16, 22, 18, 19, 22, 21, 22, 25, 24, 25, 37, 27, 28, 37, 30, 31, 37, 33, 34, 37, 36, 37, 46, 39, 40, 46, 42, 43, 46, 45, 46, 52, 48, 49, 58, 51, 52, 58, 54, 55, 58, 57, 58, 61, 60, 61, 73, 63, 64, 73, 66, 67, 70, 69, 70, 73, 72, 73

Offset: 1

Reinhard Zumkeller, May 28 2007

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangular Number
Eric Weisstein's World of Mathematics, Relatively Prime

Cf. A000217, A026741, A109007, A130335.

from math import gcd
def A130334(n):
    k, Tn, Tm = n+1, n*(n+1)//2, (n+1)*(n+2)//2
    while gcd(Tn,Tm) != 1:
        k += 1
        Tm += k
    return k # Chai Wah Wu, Sep 16 2021

a(n) > n+1 for n>1; a(n) > n+2 for n with n mod 3 = 0;

a(n) = n + A130335(n).

A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

Original entry on oeis.org

0, 1, 3, 9, 12, 25, 27, 49, 48, 81, 75, 121, 108, 169, 147, 225, 192, 289, 243, 361, 300, 441, 363, 529, 432, 625, 507, 729, 588, 841, 675, 961, 768, 1089, 867, 1225, 972, 1369, 1083, 1521, 1200, 1681, 1323, 1849, 1452, 2025, 1587, 2209, 1728, 2401, 1875, 2601, 2028, 2809, 2187, 3025

Offset: 0

Ilya Gutkovskiy, May 26 2019

Moebius transform of A076577.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000290, A007434, A016754, A026741, A033428, A076577, A308418, A309337.

a[n_] := If[OddQ[n], n^2, 3 n^2/4]; Table[a[n], {n, 0, 55}]
nmax = 55; CoefficientList[Series[x (1 + 3 x + 6 x^2 + 3 x^3 + x^4)/(1 - x^2)^3, {x, 0, nmax}], x]
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 3, 9, 12, 25}, 56]
Table[(7 - (-1)^n) n^2/8, {n, 0, 55}]

G.f.: x*(1 + 3*x + 6*x^2 + 3*x^3 + x^4)/(1 - x^2)^3.
G.f.: Sum_{k>=1} J_2(k)*x^k/(1 - x^(2*k)), where J_2() is the Jordan function (A007434).
E.g.f.: x*((4 + 3*x)*cosh(x) + (3 + 4*x)*sinh(x))/4.
Dirichlet g.f.: zeta(s-2)*(1 - 1/2^s).
a(n) = (7 - (-1)^n)*n^2/8.
a(n) = Sum_{d|n, n/d odd} J_2(d).
a(2*k+1) = A016754(k), a(2*k) = A033428(k).
Sum_{n>=1} 1/a(n) = 13*Pi^2/72 = 1.7820119057522453061...
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi^2/72 = 0.68538919452009434853...
Multiplicative with a(2^e) = 3*2^(2*e-2), and a(p^e) = p^(2*e) for odd primes p. - Amiram Eldar, Oct 26 2020
For n >= 1, n*a(n) = A309337(n) = Sum_{d divides n} (-1)^(d+1) * J(3, n/d), where the Jordan totient function J_3(n) = A059376. - Peter Bala, Jan 21 2024

A317312 Multiples of 12 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 12, 3, 24, 5, 36, 7, 48, 9, 60, 11, 72, 13, 84, 15, 96, 17, 108, 19, 120, 21, 132, 23, 144, 25, 156, 27, 168, 29, 180, 31, 192, 33, 204, 35, 216, 37, 228, 39, 240, 41, 252, 43, 264, 45, 276, 47, 288, 49, 300, 51, 312, 53, 324, 55, 336, 57, 348, 59, 360, 61, 372, 63, 384, 65, 396, 67, 408, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 16-gonal numbers (A274978).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 16-gonal numbers.

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008594 and A005408 interleaved.
Column 12 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15).
Cf. A274978.

{0}~Join~Riffle[2 Range@ # - 1, 12 Range@ #] &@ 35 (* or *)
CoefficientList[Series[x (1 + 12 x + x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 69}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 12, 3}, 70] (* Michael De Vlieger, Jul 26 2018 *)

a(2n) = 12*n, a(2n+1) = 2*n + 1.
From Michael De Vlieger, Jul 26 2018: (Start)
G.f.: x*(1 + 12*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 3*2^(e+1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 5*2^(1-s)). - Amiram Eldar, Oct 25 2023
a(n) = (7 + 5*(-1)^n)*n/2. - Aaron J Grech, Aug 20 2024

A317313 Multiples of 13 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 13, 3, 26, 5, 39, 7, 52, 9, 65, 11, 78, 13, 91, 15, 104, 17, 117, 19, 130, 21, 143, 23, 156, 25, 169, 27, 182, 29, 195, 31, 208, 33, 221, 35, 234, 37, 247, 39, 260, 41, 273, 43, 286, 45, 299, 47, 312, 49, 325, 51, 338, 53, 351, 55, 364, 57, 377, 59, 390, 61, 403, 63, 416, 65, 429, 67, 442, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 17-gonal numbers (A303305).
More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 17-gonal numbers.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008595 and A005408 interleaved.
Column 13 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16).
Cf. A303305.

Table[{13n, 2n + 1}, {n, 0, 35}] // Flatten (* or *)
CoefficientList[Series[(x^3 + 13 x^2 + x)/(x^2 - 1)^2, {x, 0, 69}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 13, 3}, 70] (* Robert G. Wilson v, Jul 26 2018 *)

a(n) = if(n%2==0, return((n/2)*13), return(n)) \\ Felix Fröhlich, Jul 26 2018

concat(0, Vec(x*(1 + 13*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 13*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 13*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 13*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 11/2^s). - Amiram Eldar, Oct 25 2023
a(n) = (15 + 11*(-1)^n)*n/4. - Aaron J Grech, Aug 20 2024

A352838 Irregular triangle read by rows: T(n, k) is the number of 2n-step closed walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

Original entry on oeis.org

1, 4, 28, 4, 232, 72, 12, 2156, 1008, 308, 48, 8, 21944, 13160, 5540, 1560, 420, 80, 20, 240280, 168780, 87192, 33628, 11964, 3636, 1200, 264, 72, 12, 2787320, 2168544, 1291220, 610232, 262612, 101976, 40376, 13720, 4900, 1512, 420, 112, 28

Offset: 0

Andrei Zabolotskii, Apr 05 2022

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give A002894.

T(n, k) is the number of words of length 2n equal to z^k in the Heisenberg group, presented as , where z=[x,y]. In particular, T(n, 0) = A307468(n).

			The table begins:
       1
       4
      28,      4
     232,     72,    12
    2156,   1008,   308,    48,     8
   21944,  13160,  5540,  1560,   420,   80,   20
  240280, 168780, 87192, 33628, 11964, 3636, 1200, 264, 72, 12
     ...
T(2, 0) = 28: the 4-step walks enclosing algebraic area 0 include 16 walks of the form "some two steps, then two steps right back" and 12 walks of the form "some step, step back, a different step, step back".
T(2, 1) = 4: the 4-step walks enclosing algebraic area 1 are the walks around each of the 4 squares touching the origin in the positive direction; cf. A334756(2, 1) = 8, which also counts walks around these squares in the negative direction.

Andrei Zabolotskii, Table of n, a(n) for n = 0..1403 (rows 0..25)
Cédric Béguin, Alain Valette and Andrzej Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, Journal of Geometry and Physics, 21 (1997), 337-356.
Li Gan, Algebraic Area of Lattice Random Walks and Exclusion Statistics, PhD thesis, Université Paris-Saclay, 2023. See Section 2.1.2, in particular Table 2.1 (divide terms in rows with nonzero A by 2 to get this table).
Stefan Mashkevich and Stéphane Ouvry, Area Distribution of Two-Dimensional Random Walks on a Square Lattice, J. Stat. Phys., 137 (2009), 71-78.
Morteza Mohammad-Noori, Enumeration of closed random walks in the square lattice according to their areas, arXiv:1012.3720 [math.CO], 2010. Published as: Morteza Mohammad-Noori, Enumeration of walks in the square lattice according to their areas, Journal of Combinatorial Mathematics and Combinatorial Computing, 91 (2014), 257-274.

Row lengths are A033638 = A002620 + 1.
Row n ends with 4 * A026741(n) for n > 0.
Row 16 is A178106.
Cf. A307468, A002894, A353091, A385672.
A334756 counts self-avoiding walks only.

z[0, 0, 0, 0] = 1;
z[-1, ] = z[, -1, _] = z[, , -1, ] = z[, , , -1] = 0;
z[m1_, m2_, l1_, l2_] := z[m1, m2, l1, l2] = Expand[z[m1, m2, l1 - 1, l2] + z[m1, m2, l1, l2 - 1] + q^(l2 - l1) z[m1 - 1, m2, l1, l2] + q^(l1 - l2) z[m1, m2 - 1, l1, l2]];
zN[n_] := Sum[z[m, m, n/2 - m, n/2 - m], {m, 0, n/2}];
walks[n_] := Module[{gf = zN[2 n], e}, e = Exponent[gf, q, Max]; CoefficientList[gf q^e, q][[e + 1 ;;]]];
Table[walks[n], {n, 0, 8}]

A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 6, 4, 10, 4, 12, 6, 8, 10, 16, 6, 18, 8, 12, 10, 22, 10, 20, 12, 19, 12, 28, 8, 30, 20, 20, 16, 24, 12, 36, 18, 24, 20, 40, 12, 42, 20, 24, 22, 46, 20, 42, 20, 32, 24, 52, 19, 40, 30, 36, 28, 58, 16, 60, 30, 36, 40, 48, 20, 66, 32, 44, 24

Offset: 1

Amiram Eldar, May 18 2025

The number of integers k from 1 to n such that the cubefree part (A360539) of gcd(n,k) is 1.

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

Cf. A005117, A036966, A078429, A360539, A254926.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), this sequence (cubefull), A384041 (exponentially odd), A384042 (5-rough).

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

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

Multiplicative with a(p^e) = (p^3-p^2+1)*p^(e-3) if e >= 3, p*(p-1) if e = 2, and p-1 otherwise.
a(n) >= A384039(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^6) = 0.62159731307414305346... .

A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 8, 10, 11, 9, 13, 14, 15, 13, 17, 16, 19, 15, 21, 22, 23, 21, 24, 26, 25, 21, 29, 30, 31, 27, 33, 34, 35, 24, 37, 38, 39, 35, 41, 42, 43, 33, 40, 46, 47, 39, 48, 48, 51, 39, 53, 50, 55, 49, 57, 58, 59, 45, 61, 62, 56, 53, 65, 66, 67, 51

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A268335.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), this sequence (exponentially odd), A384042 (5-rough).

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

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

Multiplicative with a(p^e) = ((p^2+p-1)*p^(e-1) - (-1)^e)/(p+1).
a(n) >= A000010(n), with equality if and only if n = 1.
Dirichlet g.f.: (zeta(s-1)*zeta(2*s)/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 = Product_{p prime} (1 - 1/p^2 + 1/(p^2+1)) = 0.93749428273130025078... .

A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 7, 4, 6, 5, 11, 4, 13, 7, 10, 8, 17, 6, 19, 10, 14, 11, 23, 8, 25, 13, 18, 14, 29, 10, 31, 16, 22, 17, 35, 12, 37, 19, 26, 20, 41, 14, 43, 22, 30, 23, 47, 16, 49, 25, 34, 26, 53, 18, 55, 28, 38, 29, 59, 20, 61, 31, 42, 32, 65, 22, 67, 34, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A003586, A007310, A065330, A065331, A372671.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).

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

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

Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
a(n) = A000010(A065331(n)) * A065330(n).
a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
Sum_{k=1..n} a(k) ~ n^2 / 3.

A084860 Expansion of (1 - 2x + 2x^2 - x^3)/(1 - 2x)^2.

Original entry on oeis.org

1, 2, 6, 15, 36, 84, 192, 432, 960, 2112, 4608, 9984, 21504, 46080, 98304, 208896, 442368, 933888, 1966080, 4128768, 8650752, 18087936, 37748736, 78643200, 163577856, 339738624, 704643072, 1459617792, 3019898880, 6241124352

Offset: 0

Paul Barry, Jun 12 2003

Partial sums are A084858. Partial sums of A084860.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A026741.

Cf. A139633.

CoefficientList[Series[(1-2x+2x^2-x^3)/(1-2x)^2,{x,0,50}],x]  (* Harvey P. Dale, Mar 30 2011 *)

a(0)=1, a(n+1) = 3*2^(n-2)*(n+3) - 0^n/4.
Equals binomial transform of nonzero terms of A026741: (1, 1, 3, 2, 5, 3, 7, 4, ...). - Gary W. Adamson, Apr 25 2008
Equals row sums of triangle A139633. - Gary W. Adamson, Apr 27 2008

cp's OEIS Frontend

A118392 Denominator of sum of reciprocals of first n tetrahedral numbers A000292.

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A130334 Smallest m>n such that the m-th and n-th triangular numbers are coprime.

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A308422 a(n) = n^2 if n odd, 3*n^2/4 if n even.

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A317312 Multiples of 12 and odd numbers interleaved.

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A317313 Multiples of 13 and odd numbers interleaved.

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A352838 Irregular triangle read by rows: T(n, k) is the number of 2n-step closed walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

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A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

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A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.