A239628 -id:A239628 - cp's OEIS frontend

A164073 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

1, 3, 2, 6, 4, 12, 8, 24, 16, 48, 32, 96, 64, 192, 128, 384, 256, 768, 512, 1536, 1024, 3072, 2048, 6144, 4096, 12288, 8192, 24576, 16384, 49152, 32768, 98304, 65536, 196608, 131072, 393216, 262144, 786432, 524288, 1572864, 1048576, 3145728, 2097152, 6291456

Offset: 1

Klaus Brockhaus, Aug 09 2009

Interleaving of A000079 and A007283.
Binomial transform is A048654. Second binomial transform is A111567. Third binomial transform is A081179 without initial 0. Fourth binomial transform is A164072. Fifth binomial transform is A164031.
Absolute second differences are the sequence itself. - Eric Angelini, Jul 30 2013
Least number having n - 1 Gaussian prime factors, counted with multiplicity, excluding units. See A239628 for a similar sequence. - T. D. Noe, Mar 31 2014
Writing the prime factorizations of the terms of this sequence, the exponents of prime factor 2 give the integers repeated (A004526), while the exponents of prime factor 3 give the sequence of alternating 0's and 1's (A000035). - Alonso del Arte, Nov 30 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A074323, A162255, A048654, A111567, A081179, A164072, A164031, A239628.

[ n le 2 select 2*n-1 else 2*Self(n-2): n in [1..42] ];

terms = 50; CoefficientList[Series[x * (1 + 3 * x)/(1 - 2 * x^2), {x, 0, terms}], x] (* T. D. Noe, Mar 31 2014 *)
Flatten[Table[{2^n, 3 * 2^n}, {n, 0, 31}]] (* Alonso del Arte, Nov 30 2016 *)
CoefficientList[Series[x (1 + 3 x)/(1 - 2 x^2), {x, 0, 44}], x] (* Michael De Vlieger, Dec 13 2016 *)

PARI
```
a(n) = (5 + (-1)^n) * 2^((2*n-9)\/4)
```

Vec(x*(1+3*x)/(1-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (5 + (-1)^n) * 2^(1/4 * (2*n - 1 + (-1)^n))/4.
G.f.: x*(1 + 3 * x)/(1 - 2 * x^2).
a(n) = A074323(n), n>=1.
a(n) = A162255(n-1), n>=2.
a(n) = A072946(n-2), n > 2. - R. J. Mathar, Aug 17 2009
a(n+3) = a(n + 2) * a(n + 1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = (2/3)a(n - 1) for odd n > 1, a(n) = 3a(n - 1) for even n. - Alonso del Arte, Nov 30 2016

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

cp's OEIS Frontend

A164073 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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