A350388 -id:A350388 - cp's OEIS frontend

A377990 a(n) = sigma(sigma(A350388(n))) * sigma(sigma(A350389(n))), where A350388 and A350389 are the largest unitary divisor of n that is a square, and the largest unitary divisor of n that is an exponentially odd number, respectively.

1, 4, 7, 8, 12, 28, 15, 24, 14, 39, 28, 56, 24, 60, 60, 32, 39, 56, 42, 96, 63, 91, 60, 168, 32, 96, 90, 120, 72, 195, 63, 104, 124, 120, 124, 112, 60, 168, 120, 234, 96, 252, 84, 224, 168, 195, 124, 224, 80, 128, 195, 192, 120, 360, 195, 360, 186, 234, 168, 480, 96, 252, 210, 128, 224, 403, 126, 312, 252, 403, 195

Offset: 1

Antti Karttunen, Nov 15 2024

nonn

Differs from A051027 at 52, 98, 156, 164, 245, ..., = A377991.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A051027, A350388, A350389, A351568, A351569, A377991.

A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A377990(n) = (sigma(sigma(A350388(n))) * sigma(sigma(A350389(n))));

a(n) = A051027(A350388(n)) * A051027(A350389(n)).

a(n) = sigma(A351568(n)) * sigma(A351569(n)).

A374485 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350388(i) = A350388(j) and A351569(i) = A351569(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 11, 12, 13, 13, 14, 10, 15, 16, 17, 18, 19, 13, 20, 21, 22, 23, 24, 25, 26, 18, 27, 28, 29, 28, 30, 31, 20, 32, 33, 22, 34, 35, 36, 37, 26, 28, 38, 39, 40, 26, 41, 29, 42, 26, 42, 43, 33, 20, 44, 45, 34, 46, 47, 48, 49, 50, 51, 34, 49, 26, 52, 53, 54, 55, 56, 34, 57, 43, 58, 59, 60, 48, 61, 62, 63, 42, 64, 33, 65, 66, 44, 67, 49, 42, 68

Offset: 1

Antti Karttunen, Aug 06 2024

nonn

Restricted growth sequence transform of the ordered pair [A350388(n), A351569(n)].

For all i, j >= 1: a(i) = a(j) => A000203(i) = A000203(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A350388, A350389, A351569.

Cf. also A286603, A291751.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351569(n) = sigma(A350389(n));
Aux374485(n) = [A350388(n), A351569(n)];
v374485 = rgs_transform(vector(up_to, n, Aux374485(n)));
A374485(n) = v374485[n];

A350389 a(n) is the largest unitary divisor of n that is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 8, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 24, 1, 26, 27, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71

Offset: 1

Amiram Eldar, Dec 28 2021

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

Cf. A000290, A001222, A007913, A268335, A350387, A350388, A350390.

f[p_, e_] := If[OddQ[e], p^e, 1]; 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]%2, f[i,1]^f[i,2], 1));} \\ Amiram Eldar, Sep 18 2023

from math import prod
from sympy import factorint
def A350389(n): return prod(p**e if e % 2 else 1 for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022

Multiplicative with a(p^e) = p^e if e is odd and 1 otherwise.
a(n) = n/A350388(n).
A001222(a(n)) = A350387(n).
a(n) = 1 if and only if n is a positive square (A000290 \ {0}).
a(n) = n if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ (1/2)*c*n^2, where c = Product_{p prime} (1 - p/(1+p+p^2+p^3)) = 0.7406196365...
Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1)). - Amiram Eldar, Sep 18 2023

A056624 Number of unitary square divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 08 2000

Unitary analog of A046951.

The number of exponential divisors (A322791) of n that are cubefree (A004709). - Amiram Eldar, Jun 03 2025

			n=256, it has 5 square divisors of which only 2,{1,256} are unitary, 3 divisors are not.
n=124 has 2 (1 and 4) square divisors, both of them unitary a(124) = 2.
n=108 has 12 divisors, 4 square divisors: {1,4,9,36} of which 1 and 4 are unitary, 9 and 36 are not. So a(108)=2. The largest unitary square divisor of 108 is 4 with 1 prime divisor so a(108) = 2^1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A004709, A008833, A034444, A046952, A055229, A056626, A059841, A162641, A322791, A350388.

isA056624 := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d and igcd(n/d^2, d) = 1:
a := n -> nops(select(k -> isA056624(n, k), [seq(1..n)])):  # Peter Luschny, Jun 13 2025

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f[p_, e_] := 2^(1 - Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 03 2022 *)

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, issquare(d))); \\ Michel Marcus, Jul 28 2017

from sympy import factorint
def A056624(n): return 1<Chai Wah Wu, Aug 03 2024

def is_A056624(n, d): return gcd(n, d) == d and gcd(n//d, d) == d and gcd(n//(d*d), d) == 1
def a(n): return len([k for k in range(1, n+1) if is_A056624(n, k)])
print([a(n) for n in range(1, 106)])  # Peter Luschny, Jun 13 2025

(define (A056624 n) (if (= 1 n) n (* (A000079 (A059841 (A067029 n))) (A056624 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

a(n) = 2^r, where r is the number of prime factors of the largest unitary square divisor of n.
Multiplicative with a(p^e) = 2^(1-(e mod 2)). - Vladeta Jovovic, Dec 13 2002
Dirichlet g.f.: zeta(s)*zeta(2*s)/zeta(3*s). - Werner Schulte, Apr 03 2018
Sum_{k=1..n} a(k) ~ n*Pi^2/(6*zeta(3)) + sqrt(n)*zeta(1/2)/zeta(3/2). - Vaclav Kotesovec, Feb 07 2019
a(n) = 2^A162641(n). - Amiram Eldar, Sep 26 2022
a(n) = A034444(A350388(n)). - Amiram Eldar, Sep 09 2023

More terms from Vladeta Jovovic, Dec 13 2002

A384052 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a square.

Original entry on oeis.org

1, 1, 2, 4, 4, 2, 6, 7, 9, 4, 10, 8, 12, 6, 8, 16, 16, 9, 18, 16, 12, 10, 22, 14, 25, 12, 26, 24, 28, 8, 30, 31, 20, 16, 24, 36, 36, 18, 24, 28, 40, 12, 42, 40, 36, 22, 46, 32, 49, 25, 32, 48, 52, 26, 40, 42, 36, 28, 58, 32, 60, 30, 54, 64, 48, 20, 66, 64, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A206369.
Cf. A013662, A350388, A350389, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), this sequence (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[OddQ[e], 1, 0]; 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]^f[i,2] - if(f[i,2]%2, 1, 0));}

Multiplicative with a(p^e) = p^e-1 if e is odd, and p^e if e is even.
a(n) = n * A047994(n) / A384054(n).
a(n) = A047994(A350389(n)) * A350388(n).
Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.74061963657217328604... .

A384054 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is an exponentially odd number.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 8, 8, 10, 11, 9, 13, 14, 15, 15, 17, 16, 19, 15, 21, 22, 23, 24, 24, 26, 27, 21, 29, 30, 31, 32, 33, 34, 35, 24, 37, 38, 39, 40, 41, 42, 43, 33, 40, 46, 47, 45, 48, 48, 51, 39, 53, 54, 55, 56, 57, 58, 59, 45, 61, 62, 56, 63, 65, 66, 67, 51

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A384041.
Cf. A013662, A268335, A350388, A350389, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), this sequence (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^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~, f[i,1]^f[i,2] - if(f[i,2]%2, 0, 1));}

from math import prod
from sympy import factorint
def A384054(n): return prod(p**e-(e&1^1) for p,e in factorint(n).items()) # Chai Wah Wu, May 21 2025

Multiplicative with a(p^e) = p^e if e is odd, and p^e-1 if e is even.
a(n) = n * A047994(n) / A384052(n).
a(n) = A047994(A350388(n)) * A350389(n).
Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 0.95692470821076622881... .

A358347 a(n) is the sum of the unitary divisors of n that are squares.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 1, 26, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 50, 1, 1, 1, 1, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 10, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1

Offset: 1

Amiram Eldar, Nov 11 2022

The number of unitary divisors of n that are squares is A056624(n).

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

Cf. A056624, A077610, A078434, A247041, A268335, A350388, A351568.

Similar sequences: A033634, A034448, A035316, A358346.

f[p_, e_] := If[OddQ[e], 1, p^e + 1]; 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]%2, 1, f[i,1]^f[i,2] + 1));}

a(n) >= 1 with equality if and only if n is an exponentially odd number (A268335).
Multiplicative with a(p^e) = p^e + 1 if e is even, and 1 otherwise.
a(n) = A034448(n)/A358346(n).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/(3*zeta(5/2)) = 0.6491241554... .
Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(3*s-2). - Amiram Eldar, Jan 29 2023
a(n) = A034448(A350388(n)). - Amiram Eldar, Sep 09 2023

A071974 Numerator of rational number i/j such that Sagher map sends i/j to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 7, 3, 10, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jun 19 2002

The Sagher map sends Product p_i^e_i / Product q_i^f_i (p_i and q_i being distinct primes) to Product p_i^(2e_i) * Product q_i^(2f_i-1). This is multiplicative.

			The Sagher map sends the following fractions to 1, 2, 3, 4, ...: 1/1, 1/2, 1/3, 2/1, 1/5, 1/6, 1/7, 1/4, 3/1, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David M. Bradley, Counting the Positive Rationals: A Brief Survey, arXiv:math/0509025 [math.HO], 2005.
Gerald Freilich, A denumerability formula for the rationals, Amer. Math. Monthly, Nov 1965, pp. 1013-1014.
Kevin McCrimmon, Enumeration of the positive rationals, Amer. Math. Monthly, Nov 1960, p. 868.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
Index entries for sequences related to enumerating the rationals

Cf. A071975. Differs from A056622 at a(32).
Cf. A000290, A027748, A077591, A124010, A350388.
For other bijective mappings from integers to positive rationals see A002487, A020652/A020653, A038568/A038569, A229994/A077610, A295515.
Cf. A307868.

a071974 n = product $ zipWith (^) (a027748_row n) $
   map (\e -> (1 - e `mod` 2) * e `div` 2) $ a124010_row n
-- Reinhard Zumkeller, Jun 15 2012

f[{p_, a_}] := If[EvenQ[a], p^(a/2), 1]; a[n_] := Times@@(f/@FactorInteger[n])
Table[Sqrt@ SelectFirst[Reverse@ Divisors@ n, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 104}] (* Michael De Vlieger, Dec 06 2018 *)

a(n)=local(v=factor(n)~); prod(k=1,length(v),if(v[2,k]%2,1,v[1,k]^(v[2,k]/2)))

from math import prod
from sympy import factorint
def A071974(n): return prod(p**(e>>1) for p, e in factorint(n).items() if e&1^1) # Chai Wah Wu, Jul 27 2024

If n=Product p_i^e_i, then a_n=Product p_i^f(e_i), where f(n)=n/2 if n is even and f(n)=0 if n is odd. - Reiner Martin, Jul 08 2002
a(n^2) = n, A071975(n^2) = 1, cf. A000290; a(2*(2*n-1)^2) = 2*n+1, A071975(2*(2*n-1)^2) = 2, cf. A077591. - Reinhard Zumkeller, Jul 10 2011
From Amiram Eldar, Nov 02 2023, Jul 26 2024: (Start)
a(n) = sqrt(A350388(n)) (square root of largest unitary divisor of n that is a square).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s) - 1/p^(3*s-1)). (End)
From Vaclav Kotesovec, May 05 2025: (Start)
Let f(s) = Product_{p prime} (1 -  (p^s + p)/((p^s + 1)*p^(2*s))).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A307868 = Product_{p prime} (1 - 2/(p*(1+p))) = 0.4716806136129978680752356330804820874259263820069868836357372554177321...
f'(1) = f(1) * Sum_{p prime} (5*p+3)*log(p) / ((p+1)*(p^2+p-2)) = f(1) * 2.1244279471327068377850377690765768532203174482128717024402373817115555...
and gamma is the Euler-Mascheroni constant A001620. (End)

More terms from Reiner Martin, Jul 08 2002

Additional references supplied by Kevin Ryde added by N. J. A. Sloane, May 31 2012

A351568 Sum of the divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 1, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 1, 31, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 91, 1, 1, 1, 1, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 13, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 1, 1, 13, 1, 7, 1, 1, 1, 1, 1, 57, 13

Offset: 1

Antti Karttunen, Feb 23 2022

Obviously, all terms are odd.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n).

Cf. A000203, A002117, A350388, A351569, A351570, A351575 (positions of primes).

f[p_, e_] := If[EvenQ[e], (p^(e + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)

A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351568(n) = sigma(A350388(n));

from math import prod
from sympy import factorint
def A351568(n): return prod(1 if e % 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is even and 1 otherwise.
a(n) = A000203(A350388(n)).
a(n) = A000203(n) / A351569(n).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) + 1/p^2 - 1/p^(5/2)) = 1.008259499413... . - Amiram Eldar, Nov 20 2022
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 03 2023

A367168 The largest unitary divisor of n that is a term of A138302.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Nov 07 2023

First differs from A056192 at n = 32 and from A270418 at n = 128.

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

Cf. A048298, A065465, A138302, A367169, A367170, A367171.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A360539.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^e, 1]; 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 << valuation(f[i, 2], 2), f[i, 1]^f[i, 2], 1));}

from math import prod
from sympy import factorint
def A367168(n): return prod(p**e for p,e in factorint(n).items() if not(e&-e)^e) # Chai Wah Wu, Nov 10 2023

Multiplicative with a(p^e) = p^A048298(e).
a(n) <= n, with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.881513... (A065465).

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A377990 a(n) = sigma(sigma(A350388(n))) * sigma(sigma(A350389(n))), where A350388 and A350389 are the largest unitary divisor of n that is a square, and the largest unitary divisor of n that is an exponentially odd number, respectively.

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A374485 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350388(i) = A350388(j) and A351569(i) = A351569(j), for all i, j >= 1.

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A350389 a(n) is the largest unitary divisor of n that is an exponentially odd number (A268335).

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A056624 Number of unitary square divisors of n.

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A384052 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a square.

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A384054 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is an exponentially odd number.

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A358347 a(n) is the sum of the unitary divisors of n that are squares.

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