A086275 -id:A086275 - cp's OEIS frontend

A332476 The number of unitary divisors of n in Gaussian integers.

1, 2, 2, 2, 4, 4, 2, 2, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 2, 8, 4, 4, 2, 4, 4, 8, 2, 4, 4, 16, 2, 2, 4, 8, 8, 4, 4, 4, 8, 8, 4, 8, 2, 4, 8, 4, 2, 4, 2, 8, 8, 8, 4, 4, 8, 4, 4, 8, 2, 16, 4, 4, 4, 2, 16, 8, 2, 8, 4, 16, 2, 4, 4, 8, 8, 4, 4, 16, 2, 8, 2, 8, 2, 8, 16

Offset: 1

Amiram Eldar, Feb 13 2020

			a(2) = 2 since 2 = -i * (1 + i)^2, so it has 2 unitary divisors (up to associates): 1 and (1 + i)^2.

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

Cf. A034444, A062327, A086275, A332472, A332473, A332474.

f[p_, e_] := If[Abs[p] == 1, 1, 2]; a[n_] := Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]; Array[a, 100]

Multiplicative with a(p^e) = 4 if p == 1 (mod 4) and 2 otherwise.

a(n) = 2^A086275(n).

A239629 Factored over the Gaussian integers, the least positive number having n prime factors, including units -1, i, and -i.

Original entry on oeis.org

1, 2, 5, 10, 30, 130, 390, 2730, 13260, 64090, 192270, 1345890, 7113990, 49797930, 291673590, 2041715130

Offset: 1

T. D. Noe, Mar 31 2014

Here -1, i, and -i are counted as factors. The factor 1 is counted only in a(1).

Indices of records of A239627. - Amiram Eldar, Jun 27 2020

Cf. A001221, A001222 (integer factorizations).
Cf. A078458, A086275 (Gaussian factorizations).
Cf. A239627 (number of Gaussian factors of n, including units).
Cf. A239628 (similar to this sequence, but count all prime factors).
Cf. A239630 (number of distinct factors, excluding units).

nn = 12; t = Table[0, {nn}]; n = 0; found = 0; While[found < nn, n++; cnt = Length[FactorInteger[n, GaussianIntegers -> True]]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; found++]]; t

a(14)-a(16) from Amiram Eldar, Jun 27 2020

A239630 Factored over the Gaussian integers, the least number having n prime factors, excluding units 1, -1, i, and -i.

Original entry on oeis.org

2, 5, 10, 30, 130, 390, 2210, 6630, 46410, 192270, 1345890, 7113990, 49797930, 291673590, 2041715130

Offset: 1

T. D. Noe, Mar 31 2014

From Amiram Eldar, Jun 27 2020: (Start)
Indices of records of A086275.
Also, numbers with a record number of unitary divisors in Gaussian integers (A332476). (End)

Cf. A001221, A001222 (integer factorizations).
Cf. A078458, A086275 (Gaussian factorizations).
Cf. A239627 (number of Gaussian factors of n, including units).
Cf. A239628 (similar to this sequence, but count all prime factors).
Cf. A239629 (number of distinct factors, including units).
Cf. A332476.

nn = 12; t = Table[0, {nn}]; n = 0; found = 0; While[found < nn, n++; f = FactorInteger[n, GaussianIntegers -> True]; cnt = Length[f]; If[MemberQ[{-1, I, -I}, f[[1, 1]]], cnt--]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; found++]]; t

a(13)-a(15) from Amiram Eldar, Jun 27 2020

A376645 The maximum exponent in the factorization of n into powers of Gaussian primes.

Original entry on oeis.org

0, 2, 1, 4, 1, 2, 1, 6, 2, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 6, 2, 2, 3, 4, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 2, 2, 1, 8, 2, 2, 1, 4, 1, 3, 1, 6, 1, 2, 1, 4, 1, 2, 2, 12, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 2, 4, 1, 2, 1, 8, 4, 2, 1, 4, 1, 2, 1

Offset: 1

Amiram Eldar, Oct 01 2024

a(n) = 0 only for n = 1. a(n) = k occurs infinitely many times for k >= 1. The numbers n = 2^e * m = 2^A007814(n) * A000265(n) for which a(n) = k and their asymptotic density are as follows:
  1. k = 1: n is an odd squarefree number (A056911) and the density is d(1) = 2/(3*zeta(2)) = 0.405284... (A185199).
  2. k >= 3 is odd: e < (k+1)/2 and m is a (k+1)-free number that is not a k-free number: d(k) = (1 - 1/2^((k+1)/2)) * (f(k+1)/zeta(k+1) - f(k)/zeta(k)), where f(k) = 1 - 1/2^k.
  3. k >= 2 is even: e = k/2 and m is a (k+1)-free number, or e < k/2 and m is a (k+1)-free number that is not a k-free number: d(k) = (1/2^(k/2+1)) * f(k+1)/zeta(k+1) + (1-1/2^(k/2)) * (f(k+1)/zeta(k+1) - f(k)/zeta(k)), where f(k) is defined above.
The asymptotic mean of this sequence is Sum_{k>=1} k * d(k) = 2.64836785173193409440576... .

			a(2) = 2 because 2 = -i * (1+i)^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gaussian Prime.
Wikipedia, Gaussian integer: Unique factorization.

Cf. A000265, A051903, A007814, A056911, A078458, A086275, A185199, A335852, A375669.

a[n_] := Max[FactorInteger[n, GaussianIntegers -> True][[;; , 2]]]; a[1] = 0; Array[a, 100]
(* or *)
a[n_] := Module[{e = IntegerExponent[n, 2], od, em}, odd = n / 2^e; Max[2*e, If[odd == 1, 0, Max[FactorInteger[odd][[;;, 2]]]]]]; Array[a, 100]

a(n) = if(n == 1, 0, vecmax(factor(n*I)[, 2]));

a(n) = my(e = valuation(n, 2), es = factor(n >> e)[, 2]); max(2*e, if(#es, vecmax(es), 0));

a(n) = max(2*A007814(n), A051903(A000265(n))) = max(2*A007814(n), A375669(n)).

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A332476 The number of unitary divisors of n in Gaussian integers.

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A239629 Factored over the Gaussian integers, the least positive number having n prime factors, including units -1, i, and -i.

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Extensions

A239630 Factored over the Gaussian integers, the least number having n prime factors, excluding units 1, -1, i, and -i.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A376645 The maximum exponent in the factorization of n into powers of Gaussian primes.

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