A061704 -id:A061704 - cp's OEIS frontend

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A056170(n) if A056170(n) is positive, or 1 if A056170(n) = 0.
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/A000005(n) and A037445(n)/A000005(n).
For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k.  These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796.
Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).

			First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.

Jason Kimberley, Table of i, a(i) for i = 1..10575 (n = 1..10000)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

A181800 gives first integer of each second signature. Also see A212171, A212173-A212181, A212642-A212644.
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181.
Additive: A046660, A056170.
Other: A007424, A051903 (for n > 1), A056626, A066301, A071325, A072411, A091050, A107078, A185102 (for n > 1), A212180.
Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).

&cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012

row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)

For nonsquarefree n, row n is identical to row A057521(n) of table A212171.

A053150 Cube root of largest cube dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Henry Bottomley, Feb 28 2000

This can be thought as a "lower 3rd root" of a positive integer. Upper k-th roots were studied by Broughan (2002, 2003, 2006). The sequence of "upper 3rd root" of positive integers is given by A019555. - Petros Hadjicostas, Sep 15 2019

Antti Karttunen, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
Vaclav Kotesovec, Graph - the asymptotic ratio.

Cf. A000188 (inner square root), A019554 (outer square root), A019555 (outer third root), A053164 (inner 4th root), A053166 (outer 4th root), A015052 (outer 5th root), A015053 (outer 6th root).

Cf. A000189, A000190, A008834, A008835, A015051, A061704.

f[list_] := list[[1]]^Quotient[list[[2]], 3]; Table[Apply[Times, Map[f,FactorInteger[n]]], {n, 1, 81}] (* Geoffrey Critzer, Jan 21 2015 *)
Table[SelectFirst[Reverse@ Divisors@ n, IntegerQ[#^(1/3)] &]^(1/3), {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f[p_, e_] := p^Floor[e/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

A053150(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= (f[k, 1]^(f[k, 2]\3)); ); m; } \\ Antti Karttunen, Jul 28 2017

a(n) = my(f = factor(n)); for (k=1, #f~, f[k,2] = f[k,2]\3); factorback(f); \\ Michel Marcus, Jul 28 2017

from math import prod
from sympy import factorint
def A053150(n): return prod(p**(q//3) for p, q in factorint(n).items()) # Chai Wah Wu, Aug 18 2021

Multiplicative with a(p^e) = p^[e/3]. - Mitch Harris, Apr 19 2005
a(n) = A008834(n)^(1/3) = sqrt(A000189(n)/A000188(A050985(n))).
Dirichlet g.f.: zeta(3s-1)*zeta(s)/zeta(3s). - R. J. Mathar, Apr 09 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n / (6*zeta(3)) + 3*zeta(2/3) * n^(2/3) / Pi^2. - Vaclav Kotesovec, Jan 31 2019
a(n) = Sum_{d^3|n} phi(d). - Ridouane Oudra, Dec 30 2020
G.f.: Sum_{k>=1} phi(k) * x^(k^3) / (1 - x^(k^3)). - Ilya Gutkovskiy, Aug 20 2021

More terms from Antti Karttunen, Jul 28 2017

A063775 Number of 4th powers dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Aug 16 2001

			a(79) = 1 since 79 is divisible by 1 = 1^4.
a(80) = 2 since 80 is divisible by 1 and 16 = 2^4.
a(81) = 2 since 81 is divisible by 1 and 81 = 3^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harry J. Smith)

Cf. A046951 (number of squares), A061704 (number of cubes).

Cf. A000005, A053164, A046951, A000188.

seq(coeff(series(add(x^(k^4)/(1-x^(k^4)),k=1..n),x,n+1), x, n), n = 1 .. 120); # Muniru A Asiru, Dec 29 2018

nn = 100;f[list_, i_] := list[[i]];
Table[DirichletConvolve[f[Boole[Map[IntegerQ[#] &, Map[#^(1/4) &,Range[nn]]]], n],f[Table[1, {nn}], n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 07 2015 *)
f[p_, e_] := 1 + Floor[e/4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

{ for (n=1, 2000, k=2; a=1; while ((p=k^4) <= n, if (n%p == 0, a++); k++); write("b063775.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 30 2009

a(n) = A000005(A053164(n)) = A046951(A000188(n)).
Multiplicative with a(p^e) = 1 + floor(e/4).
Dirichlet g.f.: zeta^2(4s)*Product_{primes p} (1 + p^(-s) + p^(-2s) + p^(-3s)). - R. J. Mathar, Jan 11 2012
G.f.: Sum_{k>=1} x^(k^4)/(1 - x^(k^4)). - Ilya Gutkovskiy, Mar 21 2017
Dirichlet g.f.: zeta(s) * zeta(4s). - Álvar Ibeas, Dec 29 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n / 90 + Zeta(1/4) * n^(1/4). - Vaclav Kotesovec, Feb 03 2019

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

A327626 Expansion of Sum_{k>=1} x^(k^3) / (1 - x^(k^3))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 18, 19, 20, 21, 22, 23, 27, 25, 26, 28, 28, 29, 30, 31, 36, 33, 34, 35, 36, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 54, 49, 50, 51, 52, 53, 56, 55, 63, 57, 58, 59, 60, 61, 62, 63, 73, 65, 66, 67, 68, 69, 70, 71, 81, 73, 74, 75

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

Sum of divisors d of n such that n/d is a cube.

Inverse Moebius transform of A078429.

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

Cf. A000578, A004709 (fixed points), A010057, A061704, A076752, A078429, A113061.

nmax = 75; CoefficientList[Series[Sum[x^(k^3)/(1 - x^(k^3))^2, {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[(n/#)^(1/3)] &]; Table[a[n], {n, 1, 75}]
f[p_, e_] := (p^(e+3) - p^Mod[e, 3])/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

A327626(n) = sumdiv(n,d,ispower(n/d,3)*d); \\ Antti Karttunen, Sep 19 2019

a(n) = Sum_{d|n} A078429(d).
a(n) = Sum_{d|n} A010057(n/d) * d. Dirichlet convolution of A000027 and A010057.
D.g.f.: zeta(s-1)*zeta(3s). - R. J. Mathar, Jun 05 2020
Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1890. - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = (p^(e+3) - p^(e mod 3))/(p^3-1). - Amiram Eldar, May 25 2025

A013937 a(n) = Sum_{k=1..n} floor(n/k^3).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 0

N. J. A. Sloane, Henri Lifchitz

nonn

			a(36) = [36/1]+[36/8]+[36/27]+[36/64]+... = 36+4+1+0+... = 41.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Benoit Cloitre, Plot of (a(n)-zeta(3)*n)/n^(1/3)-zeta(1/3)

Cf. A005187, A006218, A011371, A013936, A013939 for similar sequences.

A013937:=n->add(floor(n/k^3), k=1..n); seq(A013937(n), n=0..100); # Wesley Ivan Hurt, Feb 15 2014

Table[Sum[Floor[n/k^3], {k, n}], {n, 0, 100}] (* Wesley Ivan Hurt, Feb 15 2014 *)

a(n)=sum(k=1,ceil(n^(1/3)),n\k^3) /*Benoit Cloitre, Nov 05 2012 */

a(n) = a(n-1)+A061704(n). a(n) = Sum_{k=1..n} floor((n/k)^(1/3)) with asymptotic formula: a(n) = zeta(3)*n+zeta(1/3)*n^(1/3)+O(n^theta) where theta<1/3 and we conjecture that theta=1/4+epsilon is the best possible choice. - Benoit Cloitre, Nov 05 2012

G.f.: (1/(1 - x))*Sum_{k>=1} x^(k^3)/(1 - x^(k^3)). - Ilya Gutkovskiy, Feb 11 2017

More terms from Henry Bottomley, Jul 03 2001

A279495 Number of tetrahedral numbers dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5

Offset: 1

Ilya Gutkovskiy, Dec 13 2016

Inverse Möbius transform of A023533. - Antti Karttunen, Oct 01 2018

Records are a(1) = 1, a(4) = 2, a(20) = 4, a(120) = 5, a(280) = 6, a(560) = 7, a(840) = 8, a(1680) = 9, a(9240) = 11, a(18480) = 12, a(55440) = 13, a(120120) = 14, a(240240) = 15, a(314160) = 16, a(628320) = 17, a(1441440) = 18, a(2282280) = 19, a(4564560) = 21, a(9129120) = 22, a(13693680) = 23, a(27387360) = 24, a(54774720) = 25, a(68468400) = 26, a(77597520) = 27, a(136936800) = 28, a(155195040) = 29, a(310390080) = 30, a(465585120) = 31, a(775975200) = 32, a(1163962800) = 33, a(2327925600) = 36, a(4655851200) = 37, a(13967553600) = 38, a(16295479200) = 40. - Charles R Greathouse IV, Dec 19 2016

			a(10) = 2 because 10 has 4 divisors {1,2,5,10} among which 2 divisors {1,10} are tetrahedral numbers.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Tetrahedral Number.
Index to sequences related to pyramidal numbers.

Cf. A000292, A007862, A023533, A061704.

Table[SeriesCoefficient[Sum[x^(k (k + 1) (k + 2)/6)/(1 - x^(k (k + 1) (k + 2)/6)), {k, 1, n}], {x, 0, n}], {n, 1, 120}]

a(n)=sum(k=1,sqrtnint(6*n,3),n%(k*(k+1)*(k+2)/6)==0) \\ Charles R Greathouse IV, Dec 13 2016

isA000292(n)=my(k=sqrtnint(6*n,3)); k*(k+1)*(k+2)==6*n
a(n)=sumdiv(n,d,isA000292(d)) \\ Charles R Greathouse IV, Dec 13 2016

G.f.: Sum_{k>=1} x^(k*(k+1)*(k+2)/6)/(1 - x^(k*(k+1)*(k+2)/6)).
a(n) = Sum_{d|n} A023533(d). - Antti Karttunen, Oct 01 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Jan 02 2024

A294875 a(n) = Product_{d|n, d = x^k, with x,k > 1} prime(A052409(d)-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 2, 1, 1, 1, 30, 1, 2, 1, 2, 1, 1, 1, 6, 2, 1, 6, 2, 1, 1, 1, 210, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 30, 2, 2, 1, 2, 1, 6, 1, 6, 1, 1, 1, 2, 1, 1, 2, 2310, 1, 1, 1, 2, 1, 1, 1, 24, 1, 1, 2, 2, 1, 1, 1, 30, 30, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 210, 1, 2, 2, 8, 1, 1, 1, 6, 1, 1, 1, 24, 1, 1, 1, 30, 1, 1, 1, 2

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

For all i, j:
  a(i) = a(j) => A294874(i) = A294874(j) => A046951(i) = A046951(j).
  a(i) = a(j) => A061704(i) = A061704(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A001597, A052409.
Cf. A293524, A294873, A294874.
Cf. A046951, A061704, A091050 (some of the matched sequences).

A294875(n) = { my(m=1,e); fordiv(n,d, if(d>1, e = ispower(d); if(e>1, m *= prime(e-1)))); m; };

a(n) = Product_{d|n, d>1} A008578(A052409(d)).
a(n) = A064989(A293524(n)).
Other identities. For all n >= 1:
1 + A001222(a(n)) = A091050(n).

A362852 The number of divisors of n that are both bi-unitary and exponential.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 05 2023

First differs from A061704 at n = 128, and from A304327 and abs(A307428) at n = 64.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a bi-unitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e but k != e/2.
The least term that is higher than 2 is a(64) = 3.
This sequence is unbounded. E.g., a(2^(2^prime(n))) = prime(n).

			a(8) = 2 since 8 has 2 divisors that are both bi-unitary and exponential: 2 and 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biunitary Divisor.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A000005, A004709, A049419, A188999, A222266, A286324, A322791, A359411, A362853 (indices of records), A362854.

Cf. A061704, A304327, A307428.

f[p_, e_] := DivisorSigma[0, e] - If[OddQ[e], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i, 2]) - !(f[i, 2] % 2));}

Multiplicative with a(p^e) = d(e) if e is odd, and d(e)-1 if e is even, where d(k) is the number of divisors of k (A000005).
a(n) = 1 if and only if n is cubefree (A004709).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (d(k)+(k mod 2)-1)/p^k) = 1.1951330849... .

A368248 The number of unitary divisors of the cubefull part of n (A360540).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 19 2023

First differs from A061704 and A362852 at n = 64, and from A304327 at n = 72.
Also, the number of squarefree divisors of the cubefull part of n.
Also, the number of cubes of squarefree numbers (A062838) that divide n.
The number of unitary divisors of n that are cubefull numbers (A036966). - Amiram Eldar, Jun 19 2025

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

Cf. A004709, A034444, A036966, A062838, A157289, A307428, A323308, A360540, A366145.

Cf. A061704, A304327, A362852.

f[p_, e_] := If[e > 2, 2, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x < 3, 1, 2), factor(n)[, 2]));

a(n) = A034444(A360540(n)).
a(n) = abs(A307428(n)).
Multiplicative with a(p) = 1 for e <= 2, and 2 for e >= 3.
a(n) >= 1, with equality if and only if n is cubefree (A004709).
a(n) <= A034444(n), with equality if and only if n is cubefull (A036966).
Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(6*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(6) = 1.181564... (A157289).
In general, the asymptotic mean of the number of unitary divisors of the k-full part of n is zeta(k)/zeta(2*k).

cp's OEIS Frontend

A212172 Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.

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A053150 Cube root of largest cube dividing n.

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A063775 Number of 4th powers dividing n.

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

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A327626 Expansion of Sum_{k>=1} x^(k^3) / (1 - x^(k^3))^2.

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A013937 a(n) = Sum_{k=1..n} floor(n/k^3).

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A279495 Number of tetrahedral numbers dividing n.

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