cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A078910 Let r+i*s be the sum of the distinct first-quadrant Gaussian integers dividing n; sequence gives r values.

Original entry on oeis.org

1, 4, 4, 10, 9, 16, 8, 22, 13, 37, 12, 40, 19, 32, 36, 46, 23, 52, 20, 93, 32, 48, 24, 88, 56, 77, 40, 80, 37, 148, 32, 94, 48, 95, 72, 130, 45, 80, 76, 205, 51, 128, 44, 120, 117, 96, 48, 184, 57, 231, 92, 193, 63, 160, 108, 176, 80, 151, 60, 372, 73, 128, 104, 190, 176
Offset: 1

Views

Author

N. J. A. Sloane, Jan 11 2003

Keywords

Comments

A Gaussian integer z = x+iy is in the first quadrant if x > 0, y >= 0. Just one of the 4 associates z, -z, i*z, -i*z is in the first quadrant.

Examples

			The distinct first-quadrant divisors of 4 are 1, 1+i, 2, 2+2*i, 4, with sum 10+3*i, so a(4) = 10.

Links

Crossrefs

Cf. A000203, A078911, A078458, A078908, A078909, A078930.
Cf. A062327 for the number of first quadrant divisors of n.

Programs

  • Mathematica
    Table[Re[Plus@@Divisors[n, GaussianIntegers -> True]], {n, 65}] (* Alonso del Arte, Jan 24 2012 *)
  • PARI
    A078910(n,S=[])=sigma(n)+sumdiv(n*I,d,if(real(d)&imag(d)&!setsearch(S,d=vecsort(abs([real(d),imag(d)]))),S=setunion(S,[d]);(d[1]+d[2])>>(d[1]==d[2]))) \\ M. F. Hasler, Nov 22 2007

Formula

a(n) = A078911(n)+A000203(n). - Vladeta Jovovic, Jan 11 2003

Extensions

More terms from Vladeta Jovovic, Jan 11 2003

A239627 Factored over the Gaussian integers, n has a(n) distinct prime factors, including units -1, i, and -i.

Original entry on oeis.org

1, 2, 1, 2, 3, 3, 1, 2, 1, 4, 1, 3, 3, 3, 4, 1, 3, 3, 1, 4, 2, 3, 1, 3, 3, 4, 1, 3, 3, 5, 1, 2, 2, 4, 4, 3, 3, 3, 4, 3, 3, 4, 1, 3, 4, 3, 1, 2, 1, 4, 4, 4, 3, 3, 4, 3, 2, 4, 1, 5, 3, 3, 2, 2, 5, 4, 1, 4, 2, 5, 1, 3, 3, 4, 4, 3, 2, 5, 1, 4, 1, 4, 1, 4, 5, 3, 4
Offset: 1

Views

Author

T. D. Noe, Mar 31 2014

Keywords

Comments

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

Examples

			a(2) = 2 because 2 = -i * (1 + i)^2.
a(3) = 1 because 3 is prime over the complex numbers.
a(4) = 2 because 4 = -1 * (1 + i)^4.

Links

Crossrefs

Cf. A001221, A001222 (integer factorizations).
Cf. A078458, A086275 (Gaussian factorizations).
Cf. A239626 (Gaussian factorization including units).

Programs

  • Mathematica
    Table[Length[FactorInteger[n, GaussianIntegers -> True]], {n, 100}]

A239626 Factored over the Gaussian integers, n has a(n) prime factors counted multiply, including units -1, i, and -i.

Original entry on oeis.org

1, 3, 1, 5, 3, 4, 1, 7, 2, 5, 1, 6, 3, 4, 4, 8, 3, 5, 1, 7, 2, 4, 1, 8, 5, 5, 3, 6, 3, 6, 1, 11, 2, 5, 4, 7, 3, 4, 4, 8, 3, 5, 1, 6, 5, 4, 1, 9, 2, 7, 4, 7, 3, 6, 4, 8, 2, 5, 1, 8, 3, 4, 3, 13, 5, 5, 1, 7, 2, 6, 1, 9, 3, 5, 6, 6, 2, 6, 1, 11, 4, 5, 1, 7, 5, 4, 4
Offset: 1

Views

Author

T. D. Noe, Mar 31 2014

Keywords

Comments

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

Examples

			a(2) = 3 because 2 = -i * (1 + i)^2.
a(3) = 1 because 3 is prime over the complex numbers.
a(4) = 5 because 4 = -1 * (1 + i)^4.

Links

Crossrefs

Cf. A001221, A001222 (integer factorizations).
Cf. A078458, A086275 (Gaussian factorizations).
Cf. A239627 (Gaussian factorization including units).

Programs

  • Mathematica
    Table[Total[Transpose[FactorInteger[n, GaussianIntegers -> True]][[2]]], {n, 100}]

A239628 Factored over the Gaussian integers, the least positive number having n prime factors counted multiply, including units -1, i, and -i.

Original entry on oeis.org

1, 9, 2, 6, 4, 12, 8, 16, 48, 144, 32, 96, 64, 192, 128, 256, 768, 2304, 512, 1536, 1024, 3072, 2048, 4096, 12288, 36864, 8192, 24576, 16384, 49152, 32768, 65536, 196608, 589824, 131072, 393216, 262144, 786432, 524288, 1048576, 3145728, 9437184, 2097152
Offset: 1

Views

Author

T. D. Noe, Mar 31 2014

Keywords

Comments

Here -1, i, and -i are counted as factors. The factor 1 is counted only in a(1). All these numbers of products of 2^k, 3, and 9.
Similar to A164073, which gives the least integer having n prime factors (over the Gaussian integers) shifted by 1.

Examples

			a(2) = 9 because 9 = 3 * 3.
a(3) = 2 because 2 = -i * (1 + i)^2.
a(4) = 6 because 6 = -i * (1 + i)^2 * 3.

Crossrefs

Cf. A001221, A001222 (integer factorizations).
Cf. A078458, A086275 (Gaussian factorizations).
Cf. A164073 (least number having n Gaussian factors, excluding units);
Cf. A239627 (number of Gaussian factors of n, including units).
Cf. A239629, A239630 (similar, but count distinct prime factors).

Programs

  • Mathematica
    nn = 30; t = Table[0, {nn}]; n = 0; found = 0; While[found < nn, n++; cnt = Total[Transpose[FactorInteger[n, GaussianIntegers -> True]][[2]]]; If[cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; found++]]; t

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

Views

Author

T. D. Noe, Mar 31 2014

Keywords

Comments

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

Crossrefs

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).

Programs

  • Mathematica
    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

Extensions

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

Views

Author

T. D. Noe, Mar 31 2014

Keywords

Comments

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)

Crossrefs

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.

Programs

  • Mathematica
    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

Extensions

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

Views

Author

Amiram Eldar, Oct 01 2024

Keywords

Comments

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... .

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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]
  • PARI
    a(n) = if(n == 1, 0, vecmax(factor(n*I)[, 2]));
  • PARI
    a(n) = my(e = valuation(n, 2), es = factor(n >> e)[, 2]); max(2*e, if(#es, vecmax(es), 0));

Formula

a(n) = max(2*A007814(n), A051903(A000265(n))) = max(2*A007814(n), A375669(n)).
Previous Showing 11-17 of 17 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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