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.

Showing 1-5 of 5 results.

A078458 Total number of factors in a factorization of n into Gaussian primes.

Original entry on oeis.org

0, 2, 1, 4, 2, 3, 1, 6, 2, 4, 1, 5, 2, 3, 3, 8, 2, 4, 1, 6, 2, 3, 1, 7, 4, 4, 3, 5, 2, 5, 1, 10, 2, 4, 3, 6, 2, 3, 3, 8, 2, 4, 1, 5, 4, 3, 1, 9, 2, 6, 3, 6, 2, 5, 3, 7, 2, 4, 1, 7, 2, 3, 3, 12, 4, 4, 1, 6, 2, 5, 1, 8, 2, 4, 5, 5, 2, 5, 1, 10, 4, 4, 1, 6, 4, 3, 3, 7, 2, 6, 3, 5, 2, 3, 3, 11, 2, 4, 3, 8, 2, 5, 1, 8
Offset: 1

Views

Author

N. J. A. Sloane, Jan 11 2003

Keywords

Comments

a(n)+1 is also the total number of factors in a factorization of n+n*i into Gaussian primes. - Jason Kimberley, Dec 17 2011
Record high values are at a(2^k) = 2*k for k = 0, 1, 2, ... . - Bill McEachen, Oct 11 2022

Examples

			2 = (1+i)*(1-i), so a(2) = 2; 9 = 3*3, so a(9) = 2.
a(1006655265000) = a(2^3*3^2*5^4*7^5*11^3) = 3*a(2)+2*a(3)+4*a(5)+5*a(7)+3*a(11) = 3*2+2*1+4*2+5*1+3*1 = 24. - _Vladeta Jovovic_, Jan 20 2003

Links

Crossrefs

Cf. A078908-A078911, A007814, A065339, A083025.
Cf. A239626, A239627 (including units).
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), this sequence ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319444.

Programs

  • Mathematica
    Join[{0}, Table[f = FactorInteger[n, GaussianIntegers -> True]; cnt = Total[Transpose[f][[2]]]; If[MemberQ[{-1, I, -I}, f[[1, 1]]], cnt--]; cnt, {n, 2, 100}]] (* T. D. Noe, Mar 31 2014 *)
a[n_]:=PrimeOmega[n, GaussianIntegers -> True]; Array[a,104] (* Stefano Spezia, Sep 29 2024 *)
  • PARI
    a(n)=my(f=factor(n)); sum(i=1,#f~,if(f[i,1]%4==3,1,2)*f[i,2]) \\ Charles R Greathouse IV, Mar 31 2014

Formula

Fully additive with a(p)=2 if p=2 or p mod 4=1 and a(p)=1 if p mod 4=3. - Vladeta Jovovic, Jan 20 2003
a(n) depends on the number of primes of the forms 4k+1 (A083025) and 4k-1 (A065339) and on the highest power of 2 dividing n (A007814): a(n) = 2*A007814(n) + 2*A083025(n) + A065339(n). - T. D. Noe, Jul 14 2003

Extensions

More terms from Vladeta Jovovic, Jan 12 2003

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
Showing 1-5 of 5 results.

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

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