A366308 -id:A366308 - cp's OEIS frontend

A366309 The number of infinitary divisors of n that are terms of A366243.

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

Offset: 1

Amiram Eldar, Oct 06 2023

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

Cf. A037445, A139352, A366242, A366243, A366245, A366247, A366246, A366308.

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := 2^s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = if(e > 3, s(e\4)) + e%4\2 \\ after Charles R Greathouse IV at A139352
a(n) = vecprod(apply(x -> 2^s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = 2^A139352(e).
a(n) = 2^A366247(n).
a(n) = A037445(n)/A366308(n).
a(n) = A037445(A366245(n)).
a(n) >= 1, with equality if and only if n is in A366242.
a(n) <= A037445(n), with equality if and only if n is in A366243.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} 2^A139352(k)/p^k) = 1.44736831993091923328... .

A369307 The number of exponentially odd divisors d of n such that n/d is also exponentially odd.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 4, 4, 2, 4, 1, 4, 2, 2, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 4, 2, 8, 2, 2, 2, 4, 2, 4, 1, 2, 4, 2, 2, 4, 4, 4, 4, 4, 2, 4, 2, 4, 2, 3, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 2, 4, 2, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 19 2024

First differs from A366308 at n = 32.

Dirichlet convolution of A295316 with itself.

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

Cf. A000005, A005117, A062503, A268335, A295316, A322483, A365173, A366308, A369308.

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

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

for(n=1, 100, print1(direuler(p=2, n, (1 - X^2 + X)^2/(1 - X^2)^2)[n], ", ")) \\ Vaclav Kotesovec, Jan 19 2024

from math import prod
from sympy import factorint
def A369307(n): return prod(2 if e&1 else e>>1 for e in factorint(n).values()) # Chai Wah Wu, Jan 19 2024

Multiplicative with a(p^e) = 2 is e is odd, and e/2 if e is even.
a(n) >= 1, with equality if and only if n is the square of a squarefree number (A062503).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s)^2 * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s)))^2.
From Vaclav Kotesovec, Jan 19 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (p^(2*s) + p^s - 1)^2 / ((p^s + 1)^2 * p^(2*s)).
Let f(s) = Product_{p prime} (p^(2*s) + p^s - 1)^2 / ((p^s + 1)^2 * 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 - (2*p^2 + 2*p - 1) / (p^2*(p+1)^2)) = 0.49623881454854881762168565097162197963340069996226074849602334089041678...,
f'(1) = f(1) * Sum_{p prime} 2*(2*p + 1) * log(p) / ((p+1)*(p^2 + p - 1)) = f(1) * 1.49674466685934940187617305887881799198585080518913793200171026177150513...
and gamma is the Euler-Mascheroni constant A001620. (End)

A370650 Numbers whose number of infinitary divisors that are terms of A366242 is equal to the number of infinitary divisors that are terms of A366243.

Original entry on oeis.org

1, 8, 12, 18, 20, 27, 28, 44, 45, 50, 52, 63, 64, 68, 75, 76, 92, 98, 99, 116, 117, 124, 125, 144, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 216, 236, 242, 244, 245, 261, 268, 275, 279, 284, 292, 316, 324, 325, 332, 333, 338, 343, 356, 360, 363, 369, 387, 388, 400

Offset: 1

Amiram Eldar, Feb 24 2024

nonn

Numbers k such that A366308(k) = A366309(k).
Numbers k such that A366246(k) = A366247(k) = A064547(k)/2.
If k is a term, then all the numbers with the same prime signature as k are terms. The least terms with each prime signature are listed in A370651.
p^A039004(k) is a term for all primes p and all k >= 1.

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

Cf. A037445, A039004, A064547, A366242, A366243, A366246, A366247, A366308, A366309.

A370651 is a subsequence.

s1[0] = 0; s1[n_] := s1[n] = s1[Floor[n/4]] + If[OddQ[Mod[n, 4]], 1, 0]; f1[p_, e_] := s1[e]; a1[1] = 0; a1[n_] := Plus  @@ f1 @@@ FactorInteger[n];
s2[0] = 0; s2[n_] := s2[n] = s2[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f2[p_, e_] := s2[e]; a2[1] = 0; a2[n_] := Plus @@ f2 @@@ FactorInteger[n];
q[n_] := a1[n] == a2[n]; Select[Range[400], q]

cp's OEIS Frontend

A366309 The number of infinitary divisors of n that are terms of A366243.

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A369307 The number of exponentially odd divisors d of n such that n/d is also exponentially odd.

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A370650 Numbers whose number of infinitary divisors that are terms of A366242 is equal to the number of infinitary divisors that are terms of A366243.

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