A318672 -id:A318672 - cp's OEIS frontend

A318673 The 2-adic valuation of A318672.

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

Offset: 1

Antti Karttunen, Sep 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007814, A318672.

PARI
```
\\ Use the program given in A318672.
```

a(n) = A007814(A318672(n)).

A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

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, 2, 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 30 2022

First differs from A318672 and A325989 at n = 32.
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 an infinitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e and the bitwise AND of e and k is equal to k.
The least term that is higher than 2 is a(216) = 4.
The position of the first appearance of a prime p in this sequence is 2^A359081(p), if A359081(p) > -1. E.g., 2^39 = 549755813888 for p = 3, 2^175 = 4.789...*10^52 for p = 5, and 2^1275 = 6.504...*10^383 for p = 7.
This sequence is unbounded since A246600 is unbounded (see A359082).

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

Cf. A037445, A049419, A077609, A080948, A138302, A246600, A322791, A359081, A359082, A359412.

Cf. A318672, A325989.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = sumdiv(n, d, bitand(d, n)==d);
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i,2]));}

from math import prod
from sympy import divisors, factorint
def A359411(n): return prod(sum(1 for d in divisors(e,generator=True) if e|d == e) for e in factorint(n).values()) # Chai Wah Wu, Sep 01 2023

Multiplicative with a(p^e) = A246600(e).
a(n) = 1 if and only if n is in A138302.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A246600(k)/p^k) = 1.135514937... .

A380395 The number of unitary divisors of n that are cubes.

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, 1, 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, 2, 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, Jan 23 2025

First differs from A318672 at n = 64.

The sum of unitary divisors of n that are cubes is A380396(n).

			a(8) = 2 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3.
a(216) = 4 since 216 has 4 unitary divisors that are cubes, 1 = 1^3, 8 = 2^3, 27 = 3^3 and 216 = 6^3.

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

Cf. A000578, A046100, A056624, A061704, A077610, A307427, A318672, A366761, A380396, A380397.

Cf. A002117, A013662.

f[p_, e_] := 2^If[Divisible[e, 3], 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 2^if(f[i, 2]%3, 0, 1));}

a(n) = Sum_{d|n, gcd(d, n/d) = 1} [d is cube], where [] is the Iverson bracket.
Multiplicative with a(p^e) = 2 is e is divisible by 3, and 1 otherwise.
a(n) = abs(A307427(n)).
a(n) = A061704(n) - A380397(n).
a(n) >= 1, with equality if and only if n is not in A366761.
a(n) <= A061704(n), with equality if and only if n is biquadratefree (A046100).
Dirichlet g.f.: zeta(s)*zeta(3*s)/zeta(4*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3)/zeta(4) = 1.11062653532614811717... .
In general, the asymptotic mean of the number of unitary divisors of n that are m-th powers is zeta(m)/zeta(m+1), for m >= 2.

A318671 Numerators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of 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, 0, 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, 11, 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, 0, 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, -3

Offset: 1

Antti Karttunen, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A049599, A318672 (denominators).

up_to = (2^16)+1;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA049599(n) = factorback(apply(e -> (1+numdiv(e)),factor(n)[,2]));
v318671_72 = DirSqrt(vector(up_to, n, A049599(n)));
A318671(n) = numerator(v318671_72[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A049599(n) - Sum_{d|n, d>1, d 1.

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A318673 The 2-adic valuation of A318672.

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A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

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A380395 The number of unitary divisors of n that are cubes.

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Mathematica

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A318671 Numerators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula