A325837 -id:A325837 - cp's OEIS frontend

A325839 Exponentially-odd coreful highly composite numbers: numbers with record values of the number of exponentially odd coreful divisors (A325837).

1, 8, 32, 128, 512, 864, 3456, 7776, 13824, 31104, 124416, 279936, 497664, 1119744, 1990656, 3888000, 10077696, 15552000, 34992000, 62208000, 139968000, 248832000, 388800000, 559872000, 1259712000, 1555200000, 2239488000, 3499200000, 6220800000, 8957952000

Offset: 1

Amiram Eldar, Sep 07 2019

nonn

The corresponding record numbers of exponentially-odd divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 64, ... (see the link for more terms).
The even version of this sequence is A046952 which is the sequence of numbers with record number of square divisors (only even exponents, A046951).
Numbers with record values of the number of exponentially odd divisors are the same as the numbers with record values of the number of semi-unitary divisors (A322484). - Amiram Eldar, Sep 08 2023

Amiram Eldar, Table of n, a(n) for n = 1..290
Amiram Eldar, Table of n, a(n), A325837(a(n)) for n = 1..290

Cf. A033634, A046951, A046952, A268335, A322484, A325837.

Subsequence of A025487.

fun[p_,e_] := Floor[(e+1)/2]; a[n_] := Times@@(fun@@@FactorInteger[n]); am = 0; s={}; Do[a1=a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 1, 300000}]; s

Name corrected by Amiram Eldar, Sep 08 2023

A350390 a(n) is the largest exponentially odd divisor of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 28 2021

First differs from A331737 at n = 16.

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

Cf. A268335, A306071, A325837, A331737, A336643, A350389.

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

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A350390(n): return n*core(n)//prod(primefactors(n)) # Chai Wah Wu, Dec 30 2021

Multiplicative with a(p^e) = p^e if e is odd and p^(e-1) otherwise.
a(n) = n/A336643(n).
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^2*(p+1)) = 0.8073308216... (A306071).
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(2*s-1)). - Amiram Eldar, Sep 18 2023

A366902 The number of exponentially evil divisors of 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, 2, 1, 1, 1, 1, 3, 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, 4, 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, Oct 27 2023

First differs from A050361 at n = 128.
The number of divisors of n that are exponentially evil numbers (A262675), i.e., numbers having only evil (A001969) exponents in their canonical prime factorization.
The sum of these divisors is A366904(n) and the largest of them is A366906(n).

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

Cf. A001969, A004709, A050361, A159481, A262675, A366904, A366906.

Similar sequences: A325837, A353898, A365680, A366901.

f[p_, e_] := Floor[e/2] + If[OddQ[e] || OddQ[DigitCount[e + 1, 2, 1]], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = n\2 + (n%2 || hammingweight(n+1)%2); \\ after Charles R Greathouse IV at A159481
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = A159481(e).
a(n) >= 1, with equality if and only if n is a cubefree number (A004709).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} Sum_{k>=1} 1/p^A262675(k) = 1.241359937856... .

A365680 The number of exponentially squarefree divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 5, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Sep 15 2023

First differs from A252505 at n = 32.
The number of divisors of n that are exponentially squarefree numbers (A209061), i.e., numbers having only squarefree exponents in their canonical prime factorization.
The sum of these divisors is A365682(n) and the largest of them is A365683(n).

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

Cf. A000005, A013928, A046100, A209061, A252505, A365682, A365683.

Similar sequences: A325837, A353898.

f[p_, e_] := Count[Range[e], ?SquareFreeQ] + 1; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = sum(k=1, n, issquarefree(k)) + 1;
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = A013928(e+1) + 1.

a(n) <= A000005(n), with equality if and only if n is a biquadratefree number (A046100).

A366901 The number of exponentially odious divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 27 2023

First differs from A049599 and A282446 at n = 32, from A365551 at n = 64, and from A353898 at n = 128.
The number of divisors of n that are exponentially odious numbers (A270428), i.e., numbers having only odious (A000069) exponents in their canonical prime factorization.
The sum of these divisors is A366903(n) and the largest of them is A366905(n).

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

Cf. A000005, A000069, A004709, A115384, A270428, A366903, A366905.
Cf. A049599, A282446, A365551, A353898.
Similar sequences: A325837, A353898, A365680, A366902.

f[p_, e_] := Floor[e/2] + If[OddQ[e] || EvenQ[DigitCount[e + 1, 2, 1]], 1, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = 1 + n\2 + (n%2 || hammingweight(n+1)%2==0); \\ after Charles R Greathouse IV at A115384
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = A115384(e) + 1.

a(n) <= A000005(n), with equality if and only if n is a cubefree number (A004709).

A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Aug 25 2023

First differs from A252505 at n = 64.

The sum of these divisors is A365174(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A004524, A046101, A005117, A252505, A268335, A350390, A325837, A365171, A365174.

f[p_, e_] := Floor[(e + 5)/4] + Floor[(e + 6)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> (x+5)\4 + (x+6)\4, factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, (1 + X^2 - X^4)/((1 - X)^2*(1 + X^2)))[n], ", ")) \\ Vaclav Kotesovec, Jan 20 2024

Multiplicative with a(p^e) = floor((e+5)/4) + floor((e+6)/4) = A004524(e+5).
a(n) <= A000005(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) >= A034444(n), with equality if and only if n is squarefree (A005117).
a(n) == 1 (mod 2) if and only if n is a square of an exponentially odd number (i.e., a number whose prime factorization include only exponents e such that e == 2 (mod 4)).
From Vaclav Kotesovec, Jan 20 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - 1/(p^(2*s)*(1 + p^(2*s)))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/(p^2*(1 + p^2))) = 0.937494282731300250789438325050116436995101826036120273493270589183132928...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 + 2) * log(p) / (p^6 + 2*p^4 - 1) = f(1) * 0.192452062257404507109731932640803706644036700262364333369815000973104583...
and gamma is the Euler-Mascheroni constant A001620. (End)

A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A359411 and A367516 at n = 64.

Also, the number of unitary divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

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

Cf. A013661, A034444, A055076, A325837, A335275, A335988, A368167, A368169.

Cf. A359411, A367516.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, 2]; 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, 2]%2), 1, 2));}

a(n) = A034444(A368167(n)).
Multiplicative with a(p^e) = 2 if e is odd that is larger than 1, and 1 otherwise.
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 1.12560687309375943599... .

A365333 The number of exponentially odd coreful divisors of the largest square 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, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 01 2023

First differs from A043289, A053164, A063775, A203640 and A295658 at n = 64.
The number of squares dividing the largest exponentially odd divisor of n is A325837(n).
The sum of the exponentially odd divisors of the largest square dividing n is A365334(n). [corrected, Sep 08 2023]
The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n). - Amiram Eldar, Sep 08 2023

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

Cf. A008833, A046100, A046951, A325837, A365334.

Cf. A043289, A053164, A063775, A203640, A295658.

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

a(n) = vecprod(apply(x -> max(1, x\2), factor(n)[, 2]));

a(n) = A325837(A008833(n)).
a(n) = 1 if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = max(1, floor(e/2)).
Dirichlet g.f.: zeta(s) * zeta(4*s) * zeta(6*s) / zeta(12*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15015/(1382*Pi^2) = 1.100823... .

Name corrected by Amiram Eldar, Sep 08 2023

A365335 The number of exponentially odd coreful divisors of the square root of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 01 2023

First differs from A160338 at n = 64, and from A178489 at n = 65.
The number of divisors of the square root of the largest square dividing n is A046951(n).
The number of exponentially odd divisors of the square root of the largest square dividing n is A365549(n) and their sum is A365336(n). [corrected, Sep 08 2023]

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

Cf. A000188, A046951, A325837, A355265, A365336, A365549.

Cf. A160338, A178489.

f[p_, e_] := Max[1, Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> max(1, (x+2)\4), factor(n)[, 2]));

a(n) = A325837(A000188(n)).
a(n) > 1 if and only if n is a bicubeful number (A355265).
Multiplicative with a(p^e) = floor((e+2)/4).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 - 1/p^(4*s) + 1/p^(6*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.0181534831085... .

Name corrected by Amiram Eldar, Sep 08 2023

A365550 The number of square coreful divisors of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 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, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Sep 08 2023

First differs from A188585 at n = 64.

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n.

			a(16) = 2 since the coreful divisors of 16 are {2, 4, 8, 16}, and 2 of them, 4 and 16, are squares.

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

Cf. A001694, A005361 (number of coreful divisors), A046951 (number of square divisors), A325837.

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

a(n) = vecprod(apply(x -> x\2, factor(n)[, 2]));

Multiplicative with a(p^e) = floor(e/2).
a(n) > 0 if and only if n is a powerful number (A001694).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).

cp's OEIS Frontend

A325839 Exponentially-odd coreful highly composite numbers: numbers with record values of the number of exponentially odd coreful divisors (A325837).

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A350390 a(n) is the largest exponentially odd divisor of n.

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A366902 The number of exponentially evil divisors of n.

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A365680 The number of exponentially squarefree divisors of n.

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A366901 The number of exponentially odious divisors of n.

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A365173 The number of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

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A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

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A365333 The number of exponentially odd coreful divisors of the largest square dividing n.