A010057 -id:A010057 - cp's OEIS frontend

A210826 G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^3).

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

Offset: 1

Paul D. Hanna, Mar 27 2012

Compare to Liouville's function lambda (A008836) which satisfies the Lambert series identity: Sum_{n>=1} lambda(n)*x^n/(1-x^n) = Sum_{n>=1} x^(n^2).

This is a multiplicative sequence with Dirichlet g.f. zeta(3s)/zeta(s) and inverse Mobius transform in A010057. - R. J. Mathar, Mar 31 2012

			G.f.: x/(1-x) - x^2/(1-x^2) - x^3/(1-x^3) - x^5/(1-x^5) + x^6/(1-x^6) - x^7/(1-x^7) + x^8/(1-x^8) + x^10/(1-x^10) - x^11/(1-x^11) + ... + a(n)*x^n/(1-x^n) + ...
= x + x^8 + x^27 + x^64 + x^125 + x^216 + x^343 + ... + x^(n^3) + ...

Paul D. Hanna, Table of n, a(n) for n = 1..1035

Cf. A008836, A010057, A059269, A212793 (Dirichlet inverse), A219009.

Z := proc(n,k)
    local a,pf,e ;
    a := 1 ;
    for pf in ifactors(n)[2] do
        e := pf[2] ;
        if modp(e,k) = 0 then
            ;
        elif modp(e,k) = 1 then
            a := -a ;
        else
            a := 0 ;
        end if;
    end do;
    a;
end proc:
A210826 := proc(n)
    Z(n,3) ;
end proc: # R. J. Mathar, May 28 2016

Mod[Table[DivisorSigma[0, n], {n, 1, 100}], 3, -1] (* Geoffrey Critzer, Mar 19 2015 *)

{a(n) = if( n==0, 0, kronecker( -3, numdiv(n)))}; /* Michael Somos, Mar 28 2012 */

{a(n)=[0,1,-1][numdiv(n)%3+1]} /* a(n) == d(n) (mod 3) */

{a(n)=local(CUBES=sum(k=1, floor(n^(1/3)), x^(k^3))); if(n==1, 1, polcoeff(CUBES-sum(m=1, n-1, a(m)*x^m/(1-x^m+x*O(x^n))), n))}

/* Vectorized form (faster): */
{A=[1]; for(i=1, 256, print1(A[#A], ", "); A=concat(A, 0); A[#A]=polcoeff(sum(k=1, ceil((#A)^(1/3)), x^(k^3)) - sum(m=1, #A-1, A[m]*x^m/(1-x^m+x*O(x^#A))), #A)); print1(A[#A])}
{sum(n=1, #A, A[n]*x^n/(1-x^n+O(x^(#A))))} /* Verify Lambert series */

for(n=1, 100, print1(direuler(p=2, n, (1-X)/(1-X^3))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from math import prod
from sympy import factorint
def A210826(n): return prod((1,-1,0)[e%3] for e in factorint(n).values()) # Chai Wah Wu, Jun 18 2024

a(n) == d(n) (mod 3), where d(n) is the number of divisors of n;
a(n) = 0 iff the number of divisors of n is divisible by 3 (A059269),
a(n) = 1 iff d(n) == 1 (mod 3),
a(n) = -1 iff d(n) == 2 (mod 3).
Multiplicative with a(p^e) = -1 + ((e+2) mod 3). Thus the Dirichlet g.f. is indeed zeta(3s)/zeta(s). Also sumdiv(n,d,a(d))=1 iff n is a cube, else sumdiv(n,d,a(d))=0 hence Sum_{k=1..n} a(k)*floor(n/k) = floor(n^(1/3)). - Benoit Cloitre, Oct 28 2012

A280618 Expansion of (Sum_{k>=1} x^(k^3))^2.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 06 2017

nonn

Number of ways to write n as an ordered sum of two positive cubes.

			a(9) = 2 because we have [8, 1] and [1, 8].

Antti Karttunen, Table of n, a(n) for n = 0..65537
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Cf. A000578, A001235 (positions of terms > 3), A003325 (of nonzero terms), A010057, A063725, A173677.

nmax = 150; CoefficientList[Series[(Sum[x^(k^3), {k, 1, nmax}])^2, {x, 0, nmax}], x]

A010057(n) = ispower(n, 3);
A280618(n) = if(n<2, 0, sum(r=1,sqrtnint(n-1,3),A010057(n-(r^3)))); \\ Antti Karttunen, Nov 30 2021

G.f.: (Sum_{k>=1} x^(k^3))^2.

A048927 Numbers that are the sum of 5 positive cubes in exactly 2 ways.

Original entry on oeis.org

157, 220, 227, 246, 253, 260, 267, 279, 283, 286, 305, 316, 323, 342, 344, 361, 368, 377, 379, 384, 403, 410, 435, 440, 442, 468, 475, 487, 494, 501, 523, 530, 531, 549, 562, 568, 586, 592, 594, 595, 599, 602, 621, 625, 640, 647, 657, 658, 683, 703, 710

Offset: 1

Eric W. Weisstein

nonn

It appears that this sequence has 15416 terms, the last of which is 2243453. - Donovan Johnson, Jan 11 2013

From a(1) = 157 we see that c(n) = (number of ways n is the sum of 5 cubes) coincides with A010057 = characteristic function of cubes, up to n = 156. This sequence lists the numbers n for which c(n) = 2. See A003328 for c(n) > 0 and A048926 for c(n) = 1. - M. F. Hasler, Jan 04 2023

Donovan Johnson, Table of n, a(n) for n = 1..15416 (terms < 10^8)
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003328 (sums of 5 positive cubes), A025404, A048926 (sum of 5 positive cubes in exactly 1 way), A048930, A294736, A343702, A343705, A344237.

Select[ Range[ 1000], (test = Length[ Select[ PowersRepresentations[#, 5, 3], And @@ (Positive /@ #)& ] ] == 2; If[test, Print[#]]; test)& ](* Jean-François Alcover, Nov 09 2012 *)

(waycount(n,numcubes,imax)={if(numcubes==0, !n, sum(i=1,imax, waycount(n-i^3,numcubes-1,i)))}); isA048927(n)=(waycount(n,5,floor(n^(1/3)))==2); \\ Michael B. Porter, Sep 27 2009

def ways (n, left = 5, last = 1):
  a = last; a3 = a**3; c = 0
  while a3 <= n-left+1:
    if left > 1:
       c += ways(n-a3, left-1, a)
    elif a3 == n:
       c += 1
    a += 1; a3 = a**3
  return c
for n in range (1,1000): # to print this sequence
  if ways(n)==2: print(n,end=", ") # in Python2 use, e.g.: print n,
# Minor edits by M. F. Hasler, Jan 04 2023

More terms from Walter Hofmann (walterh(AT)gmx.de), Jun 01 2000

A052045 Cubes lacking the digit zero in their decimal expansion.

Original entry on oeis.org

1, 8, 27, 64, 125, 216, 343, 512, 729, 1331, 1728, 2197, 2744, 3375, 4913, 5832, 6859, 9261, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 29791, 32768, 35937, 42875, 46656, 54872, 59319, 68921, 85184, 91125, 97336, 117649, 132651, 148877

Offset: 1

Patrick De Geest, Dec 15 1999

This sequence is infinite since A052427(n)^3 is a term for all n>=0. - Amiram Eldar, Nov 23 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
Eric Weisstein's World of Mathematics, Zerofree [From _Reinhard Zumkeller_, Dec 01 2009]

Cubes: A052044, A051750, A051751, A051832, A051833, A052427.

Squares: A052040, A052041, A052042, A052043.

select(t -> not has(convert(t,base,10),0), [seq(m^3,m=1..10^3)]); # Robert Israel, Aug 24 2014

Select[Range[53]^3, DigitCount[#, 10, 0] == 0 &] (* Amiram Eldar, Nov 23 2020 *)

lista(nn) = {for (n=1, nn, if (vecmin(digits(cub=n^3)), print1(cub, ", ")););} \\ Michel Marcus, Aug 25 2014

A052045 = [n**3 for n in range(1,10**5) if not str(n**3).count('0')]
# Chai Wah Wu, Aug 24 2014

Intersection of A052382 and A000578; A168046(a(n))*A010057(a(n)) = 1. - Reinhard Zumkeller, Dec 01 2009

a(n) = A052044(n)^3. - Amiram Eldar, Nov 23 2020

A322885 Number of 3-generated Abelian groups of order n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1

Offset: 1

Álvar Ibeas, Dec 29 2018

Groups generated by fewer than 3 elements are not excluded. The number of Abelian groups with 3 invariant factors is a(n) - A046951(n).
Sum of the first three columns from A249770 (for n > 1).
Dirichlet convolution of A061704 and A010052. Dirichlet convolution of A046951 and A010057.
The number of unordered factorizations of n into biquadratefree power of primes (1 and primes, squares of primes and cubes of primes, A087797). - Amiram Eldar, Jun 12 2025

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001399, A010052, A010057, A046951, A061704, A087797, A249770.

f:= proc(n) local t;
  mul(round((t[2]+3)^2/12),t=ifactors(n)[2])
end proc:
map(f, [$1..200]); # Robert Israel, May 20 2019

a[n_] := Times @@ (Round[(# + 3)^2/12]& /@ FactorInteger[n][[All, 2]]);
Array[a, 102] (* Jean-François Alcover, Jan 02 2019 *)

a(n) = vecprod(apply(x -> round((x+3)^2/12), factor(n)[, 2])); \\ Amiram Eldar, Jun 12 2025

Multiplicative with a(p^e) = A001399(e).
Dirichlet g.f.: zeta(s) * zeta(2s) * zeta(3s).
Sum_{k=1..n} a(k) ~ Pi^2*zeta(3)*n/6 + zeta(1/2)*zeta(3/2)*sqrt(n) + zeta(1/3)*zeta(2/3)*n^(1/3). - Vaclav Kotesovec, Feb 02 2019

A340977 Number of ways to write n as an ordered sum of 4 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 4, 0, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 12, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 24

Offset: 4

Ilya Gutkovskiy, Feb 01 2021

nonn

Robert Israel, Table of n, a(n) for n = 4..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025457, A051344, A173678, A280618, A340978, A340979, A340980, A340981, A340982, A340983.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^3), j=1..iroot(n, 3))))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..100);  # Alois P. Heinz, Feb 01 2021

nmax = 100; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (Sum_{k>=1} x^(k^3))^4.

A340978 Number of ways to write n as an ordered sum of 5 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 5, 0, 5, 0, 0, 0, 0, 20, 0, 1, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 10, 0, 5, 0, 0, 0, 0, 30, 0, 0, 0, 5, 0, 0, 30, 0, 0, 0, 20, 0, 0, 10, 0, 0, 0, 30, 10, 0, 0, 0, 0, 0, 20, 20, 0, 0, 0, 20, 0, 5, 10

Offset: 5

Ilya Gutkovskiy, Feb 01 2021

nonn

Robert Israel, Table of n, a(n) for n = 5..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025458, A051344, A173679, A280618, A340977, A340979, A340980, A340981, A340982, A340983.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^3), j=1..iroot(n, 3))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..97);  # Alois P. Heinz, Feb 01 2021

nmax = 97; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (Sum_{k>=1} x^(k^3))^5.

A340979 Number of ways to write n as an ordered sum of 6 positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 6, 0, 15, 0, 0, 0, 0, 30, 0, 6, 0, 0, 0, 0, 60, 0, 1, 0, 0, 0, 0, 60, 0, 0, 0, 0, 15, 0, 30, 0, 0, 0, 0, 60, 0, 6, 0, 6, 0, 0, 90, 0, 0, 0, 30, 0, 0, 60, 0, 0, 0, 60, 20, 0, 15, 0, 0, 0, 60, 60, 0, 0, 0, 30, 0, 30, 60

Offset: 6

Ilya Gutkovskiy, Feb 01 2021

nonn

Robert Israel, Table of n, a(n) for n = 6..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A010057, A025459, A051344, A173680, A280618, A340977, A340978, A340980, A340981, A340982, A340983.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^3), j=1..iroot(n, 3))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..98);  # Alois P. Heinz, Feb 01 2021

nmax = 98; CoefficientList[Series[Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (Sum_{k>=1} x^(k^3))^6.

A025446 Number of partitions of n into 2 nonnegative cubes.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

a(1729) = 2, the first point where a value larger than 1 appears, and where this sequence differs from A373972. - Antti Karttunen, Jun 24 2024

			From _Antti Karttunen_, Jun 24 2024: (Start)
8 = 0^3 + 2^3, and as there are no other partitions of 8 into 2 nonnegative cubes, a(8) = 1.
16 = 2^3 + 2^3, and as there are no other partitions of 16 into 2 nonnegative cubes, a(16) = 1.
1729 = 1^3 + 12^3 = 9^3 + 10^3, and as there are no other partitions of 1729 into 2 nonnegative cubes, a(1729) = 2.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..100080
Wikipedia, 1729 (number)

Cf. A010057, A025455, A004999 (indices of nonzero terms), A373972 (their characteristic function).

A025446(n) = if(n<=2, 1, my(s=0, x=sqrtnint(n,3)); forstep(i=x, 0, -1, my(x3=i^3, y3=n-x3); if(y3>x3, return(s), s += ispower(y3, 3)))); \\ Antti Karttunen, Jun 24 2024

a(n) = A010057(n) + A025455(n) = A010057(n) XOR A025455(n). [The latter by Fermat's Last Theorem] - Antti Karttunen, Jun 24 2024

Data section extended up to a(126) and the secondary offset added by Antti Karttunen, Jun 24 2024

A078429 Number of integers k among 1..n for which gcd(k,n) is a cube.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 6, 4, 10, 4, 12, 6, 8, 9, 16, 6, 18, 8, 12, 10, 22, 10, 20, 12, 19, 12, 28, 8, 30, 18, 20, 16, 24, 12, 36, 18, 24, 20, 40, 12, 42, 20, 24, 22, 46, 18, 42, 20, 32, 24, 52, 19, 40, 30, 36, 28, 58, 16, 60, 30, 36, 37, 48, 20, 66, 32, 44, 24, 70, 30, 72, 36, 40, 36

Offset: 1

Vladeta Jovovic, Dec 29 2002

Daniel Suteu, Table of n, a(n) for n = 1..10000
Eckford Cohen, A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where a(n) = Psi_3(n).

Cf. A061020, A206369, A327626 (inv. Mob. Trans.).

Cf. A000010, A000027, A010057, A077834, A077947, A210826.

nn = 76; f[list_, i_] := list[[i]]; a = Table[If[IntegerQ[n^(1/3)], 1, 0], {n, 1, nn}]; b =Table[EulerPhi[n], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 25 2015 *)

a(n) = sum(k=1, n, ispower(gcd(n, k), 3)); \\ Michel Marcus, Feb 25 2015

a(n) = sumdiv(n, d, eulerphi(n/d) * ispower(d, 3)); \\ Daniel Suteu, Jun 27 2018

a(n) is multiplicative.
G.f. for a(p^n), p a prime, is given by 1/(1+x+x^2)/(1-p*x).
a(2^n) = A077947(n), a(3^n) = A077834(n).
a(p) = p-1, a(p^2) = p*(p-1), a(p^3) = p^3-p^2+1, a(p^4) = (p-1)*(p+1)*(p^2-p+1), ...
Dirichlet g.f.: zeta(s - 1)*zeta(3*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015
a(n) = Sum_{d|n, d is a perfect cube} phi(n/d), where phi(k) is the Euler totient function. Dirichlet convolution of A000010 and A010057. - Daniel Suteu, Jun 27 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 315. - Vaclav Kotesovec, Feb 07 2019
Dirichlet convolution of A000027 and A210826. - R. J. Mathar, Jun 05 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} A010057(gcd(n,k)).
a(n) = Sum_{k=1..n} A010057(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

cp's OEIS Frontend

A210826 G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^3).

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A280618 Expansion of (Sum_{k>=1} x^(k^3))^2.

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A048927 Numbers that are the sum of 5 positive cubes in exactly 2 ways.

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A052045 Cubes lacking the digit zero in their decimal expansion.

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A322885 Number of 3-generated Abelian groups of order n.

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A340977 Number of ways to write n as an ordered sum of 4 positive cubes.

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A340978 Number of ways to write n as an ordered sum of 5 positive cubes.

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