A103889 -id:A103889 - cp's OEIS frontend

A227380 Doubling the first two of every four nonnegative numbers.

0, 2, 2, 3, 8, 10, 6, 7, 16, 18, 10, 11, 24, 26, 14, 15, 32, 34, 18, 19, 40, 42, 22, 23, 48, 50, 26, 27, 56, 58, 30, 31, 64, 66, 34, 35, 72, 74, 38, 39, 80, 82, 42, 43, 88, 90, 46, 47, 96, 98, 50, 51, 104, 106, 54, 55, 112, 114

Offset: 0

Paul Curtz, Jul 09 2013

a(n) and its differences:
  0,    2,  2,   3,   8,  10,   6,   7,  16,...
  2,    0,  1,   5,   2,  -4,   1,   9,   2,...
  -2,   1,  4,  -3,  -6,   5,   8,  -7, -10,...   see A103889(n)
  3,    3, -7,  -3,  11,   3, -15,  -3,  19,...
  0,  -10,  4,  14,  -8, -18,  12,  22, -16,...
  -10, 14, 10,- 22, -10,  30,  10, -38, -10,... .
The inverse binomial transform is
b(n)=0, 2, -2, 3, 0, -10, 24, -28, 0, 72, -160, 176, 0,...
    =(0, 1, -1, 1, 0, -1, 4, -4, 0, 4, -16, 16, 0,...) * a(n).

Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Cf. A001477.

{2#[[1]],2#[[2]],#[[3]],#[[4]]}&/@Partition[Range[0,60],4]//Flatten (* or *) LinearRecurrence[{2,-3,4,-3,2,-1},{0,2,2,3,8,10},60] (* Harvey P. Dale, Dec 14 2021 *)

a(n) = n*A130658(n+2) = 2*A227316(n)/(n+1).
a(n) - a(n-4) = period 4:repeat 8, 8, 4, 4 = 4*A130658(n+2).
G.f.: (x^5 + 5*x^3 - 2*x^2 + 2*x)/((1-x)^2 * (1+x^2)^2). - Ralf Stephan, Jul 13 2013

A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

Original entry on oeis.org

2, 3, 1, 4, 6, 7, 5, 8, 10, 11, 9, 12, 14, 15, 13, 16, 18, 19, 17, 20, 22, 23, 21, 24, 26, 27, 25, 28, 30, 31, 29, 32, 34, 35, 33, 36, 38, 39, 37, 40, 42, 43, 41, 44, 46, 47, 45, 48, 50, 51, 49, 52, 54, 55, 53, 56, 58, 59, 57, 60, 62, 63, 61, 64, 66, 67, 65

Offset: 1

Guenther Schrack, Jan 18 2018

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the first and second elements, then swap the second and third elements; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Inverse: A292576.
Sequence of fixed points: A008586(n) for n > 0.
First differences: (-1)^floor(n^2/4)*A068073(n-1) for n > 0.
Subsequences:
  elements with odd index: A042963(A103889(n)) for n > 0.
  elements with even index A014601(n) for n > 0.
  odd elements: A166519(n-1) for n > 0.
  indices of odd elements: A042964(n) for n > 0.
  even elements: A005843(n) for n > 0.
  indices of even elements: A042948(n) for n > 0.
Other similar permutations: A116966, A284307, A292576.

a = [2 3 1 4];
max = 10000;    % Generation of b-file.
for n := 5:max
   a(n) = a(n-4) + 4;
end;

Nest[Append[#, #[[-4]] + 4] &, {2, 3, 1, 4}, 63] (* or *)
Array[# + ((-1)^# + ((-1)^(# (# - 1)/2)) (1 - 2 (-1)^#))/2 &, 67] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,0,0,1,-1},{2,3,1,4,6},70] (* Harvey P. Dale, Dec 12 2018 *)

for(n=1, 100, print1(n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2, ", "))

O.g.f.: (3*x^3 - 2*x^2 + x + 2)/(x^5 - x^4 - x - 1).
a(1) = 2, a(2) = 3, a(3) = 1, a(4) = 4, a(n) = a(n-4) + 4 for n > 4.
a(n) = n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2.
a(n) = n + (cos(n*Pi) - cos(n*Pi/2) + 3*sin(n*Pi/2))/2.
a(n) = 2*floor((n+1)/2) - 4*floor((n+1)/4) + floor(n/2) + 2*floor(n/4).
a(n) = n + (-1)^floor((n-1)^2/4)*A140081(n) for n > 0.
a(n) = A056699(n+1) - 1, n > 0.
a(n+2) = A168269(n+1) - a(n), n > 0.
a(n+2) = a(n) + (-1)^floor((n+1)^2/4)*A132400(n+2) for n > 0.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
First differences: periodic, (1, -2, 3, 2) repeat.
Compositions:
  a(n) = A080412(A116966(n-1)) for n > 0.
  a(n) = A284307(A256008(n)) for n > 0.
  a(A067060(n)) = A133256(n) for n > 0.
  A116966(a(n+1)-1) = A092486(n) for n >= 0.
  A056699(a(n)) = A256008(n) for n > 0.

A307613 Inverse of the permutation A307485: one odd, two even, four odd, eight even, etc; extended with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 8, 6, 9, 7, 10, 16, 11, 17, 12, 18, 13, 19, 14, 20, 15, 21, 32, 22, 33, 23, 34, 24, 35, 25, 36, 26, 37, 27, 38, 28, 39, 29, 40, 30, 41, 31, 42, 64, 43, 65, 44, 66, 45, 67, 46, 68, 47, 69, 48, 70, 49, 71, 50, 72, 51, 73, 52, 74, 53, 75, 54, 76, 55, 77

Offset: 0

M. F. Hasler, Apr 18 2019

nonn

See A307485 for further information, motivation & references.

Also, a(n) is the smallest k not yet in the sequence such that bitxor(k,a(n-1)) >= a(n-1). - Giorgos Kalogeropoulos, May 31 2019

			  Index n : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
A307485(n): 0, 1, 2, 4, 3, 5, 7, 9, 6, 8, ...
This sequence, the inverse permutation, is obtained by reading the above "from bottom to top", i.e., find the index in 2nd row, return the number above it: e.g., a(3) = 4, a(4) = 3, a(5) = 5, a(6) = 8, a(7) = 6, etc.

OEIS' Index to sequences related to permutations of the integers.

Cf. A307485 (inverse permutation), A307612 (partial sums thereof).
Cf. A103889 (odd & even swapped), A004442 (pairs reversed: n + (-1)^n).
Odd numbers: A005408. Even numbers: A005843.
Cf. A233275 (different permutation based on entangling odd & even numbers).

a[1]=1; a[n_] := a[n] = (t=1; While[BitXor[a[n-1],t] < a[n-1] || MemberQ[Array[a, n-1], t], t++]; t)
Join[{0}, Table[a[k], {k,100}]]  (* Giorgos Kalogeropoulos, May 31 2019 *)

my(A=apply(A307485,[1..99]), B=vecsort(A,,1)); for(i=1,#B,A[B[i]]==i||return(A307613=B[1..i-1]))

A121496 Run lengths of successive numbers in A068225.

Original entry on oeis.org

1, 2, 2, 1, 3, 4, 4, 3, 5, 6, 6, 5, 7, 8, 8, 7, 9, 10, 10, 9, 11, 12, 12, 11, 13, 14, 14, 13, 15, 16, 16, 15, 17, 18, 18, 17, 19, 20, 20, 19, 21, 22, 22, 21, 23, 24, 24, 23, 25, 26, 26, 25, 27, 28, 28, 27, 29, 30, 30, 29, 31, 32, 32, 31, 33, 34, 34, 33, 35, 36, 36, 35, 37, 38, 38

Offset: 1

Rick L. Shepherd, Aug 03 2006

A000027 and A103889 are bisections.

			The fifth run of successive numbers in A068225 is 8, 9, 10 with run length three so a(5) = 3.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000027, A068225, A103889.

Rest@ CoefficientList[Series[x (1 + x - x^3 + x^4)/((1 - x)^2*(1 + x) (1 + x^2)), {x, 0, 75}], x] (* Michael De Vlieger, Oct 02 2017 *)

a(n) = if(n%2==1,(n+1)/2,if(n%4==0,(n/2)-1,(n/2)+1))
for(n=1,80,print1(a(n),", "))

Vec(x*(1+x-x^3+x^4)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100)) \\ Colin Barker, Apr 08 2016

a(2*k-1) = k, a(4*k) = 2*k-1, a(4*k-2) = 2*k, for k >= 1.
From Colin Barker, Apr 08 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x*(1+x-x^3+x^4) / ((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = (2*n+1-4*cos(n*Pi/2)-cos(n*Pi))/4. - Wesley Ivan Hurt, Oct 02 2017

A123246 a(n) = PrimePi(n) + (-1)^(PrimePi(n) + 1) (cf. A000720).

Original entry on oeis.org

-1, 2, 1, 1, 4, 4, 3, 3, 3, 3, 6, 6, 5, 5, 5, 5, 8, 8, 7, 7, 7, 7, 10, 10, 10, 10, 10, 10, 9, 9, 12, 12, 12, 12, 12, 12, 11, 11, 11, 11, 14, 14, 13, 13, 13, 13, 16, 16, 16, 16, 16, 16, 15, 15, 15, 15, 15, 15, 18, 18, 17, 17, 17, 17, 17, 17, 20, 20, 20, 20, 19, 19, 22, 22, 22, 22, 22, 22, 21, 21, 21, 21, 24, 24, 24, 24, 24, 24, 23, 23, 23, 23

Offset: 1

Roger L. Bagula, Oct 07 2006

sign

Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, Dover, New York, 1978, page 134.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000720, A055670, A103889.

f[n_] = PrimePi[n] + (-1)^(PrimePi[n] + 1); Table[f[n], {n, 1, 200}]

a(n) = primepi(n) + (-1)^(primepi(n) + 1); \\ Michel Marcus, Oct 12 2018

Edited by N. J. A. Sloane, Oct 08 2006

A163975 n-th nonprime - (-1)^(n-th nonprime).

Original entry on oeis.org

-1, 2, 3, 5, 7, 10, 9, 11, 13, 16, 15, 17, 19, 22, 21, 23, 26, 25, 28, 27, 29, 31, 34, 33, 36, 35, 37, 40, 39, 41, 43, 46, 45, 47, 50, 49, 52, 51, 53, 56, 55, 58, 57, 59, 61, 64, 63, 66, 65, 67, 70, 69, 71, 73, 76, 75, 78, 77, 79, 82, 81, 83, 86, 85, 88, 87, 89, 92, 91, 94, 93

Offset: 1

Juri-Stepan Gerasimov, Aug 07 2009

			a(1)=0-(-1)^0=-1. a(2)=1-(-1)^1=2.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A141468.

A103889 := proc(n) n-(-1)^n ; end:
A141468 := proc(n) option remember ; if n <= 2 then n-1 ; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a); fi; od: fi; end:
A163975 := proc(n) A103889(A141468(n)) ; end: seq( A163975(n),n=1..120) ;

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@n]; Table[nonPrime[n] - (-1)^(nonPrime[n]), {n, 0, 50}] (* G. C. Greubel, Aug 24 2017 *)

for(n=1,1e3,if(!isprime(n), print1(n - (-1)^n", "))) \\ Charles R Greathouse IV, Jun 10 2015

a(n) = A141468(n) - (-1)^A141468(n).

a(n) = A103889(A141468(n)), n>0.

Entries checked by R. J. Mathar, Aug 29 2009

A226279 a(4n) = a(4n+2) = 2n , a(4n+1) = a(4n+3) = 2n-1.

Original entry on oeis.org

0, -1, 0, -1, 2, 1, 2, 1, 4, 3, 4, 3, 6, 5, 6, 5, 8, 7, 8, 7, 10, 9, 10, 9, 12, 11, 12, 11, 14, 13, 14, 13, 16, 15, 16, 15, 18, 17, 18, 17, 20, 19, 20, 19, 22, 21, 22, 21, 24, 23, 24, 23, 26, 25, 26, 25, 28, 27, 28, 27, 30, 29, 30, 29

Offset: 0

Paul Curtz, Jun 02 2013

a(n)=c(n) in A214297(n).
In A214297 d(n)=-1,1,1,3,1,3,3,... = mix (-A186422(2n) , A186422(2n+1)).
A214297 is the (reduced) numerator of 1/4 - 1/A061038(n).
(i.e. (1/4 -(1/0, 1/4, 1, 1/36, 1/16,...)) = -1/0, 0/1, -3/4, 2/9, 3/16,... )
1/0 is a convention.
n^2=(a(n+1)+d(n+1))^2 are the denominators.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A134967, A162330, A103889, A000290.

Table[{0, -1} + 2*Floor[n/2], {n, 0, 31}] // Flatten (* Jean-François Alcover, Jun 03 2013 *)

a(n)=n\4*2-n%2 \\ Charles R Greathouse IV, Sep 15 2013

a(0) = a(2)=0, a(1)=a(3)=-1, a(4)=2.
a(n) = a(n-4) + 2, n > 3.
a(n) = a(n-1) + a(n-4) - a(n-5), n > 4.
A214297(n) = a(n+1) * d(n+1).
G.f.: x*(3*x^3-x^2+x-1) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Sep 22 2013

A342769 Pairwise listing of the partitions of 2k into two parts, (s,t), with 0 < s <= t ordered by increasing values of s and where k = 1,2,... .

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 1, 5, 2, 4, 3, 3, 1, 7, 2, 6, 3, 5, 4, 4, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 1, 11, 2, 10, 3, 9, 4, 8, 5, 7, 6, 6, 1, 13, 2, 12, 3, 11, 4, 10, 5, 9, 6, 8, 7, 7, 1, 15, 2, 14, 3, 13, 4, 12, 5, 11, 6, 10, 7, 9, 8, 8, 1, 17, 2, 16, 3, 15, 4, 14, 5, 13, 6, 12, 7

Offset: 1

Wesley Ivan Hurt, Mar 21 2021

			                                                        [1,13]
                                               [1,11]   [2,12]
                                       [1,9]   [2,10]   [3,11]
                               [1,7]   [2,8]   [3, 9]   [4,10]
                       [1,5]   [2,6]   [3,7]   [4, 8]   [5, 9]
               [1,3]   [2,4]   [3,5]   [4,6]   [5, 7]   [6, 8]
       [1,1]   [2,2]   [3,3]   [4,4]   [5,5]   [6, 6]   [7, 7]
   2k    2       4       6       8       10      12       14
  --------------------------------------------------------------------------
   2k   Nondecreasing partitions of 2k
  --------------------------------------------------------------------------
   2   1,1
   4   1,3,2,2
   6   1,5,2,4,3,3
   8   1,7,2,6,3,5,4,4
  10   1,9,2,8,3,7,4,6,5,5
  12   1,11,2,10,3,9,4,8,5,7,6,6
  14   1,13,2,12,3,11,4,10,5,9,6,8,7,7
  ...

Index entries for sequences related to partitions

Cf. A000194, A052928, A103889, A339399, A339443, A342913.

a(n) = k + (k^2 + k - m)*(-1)^n / 2, where k = round(sqrt(m)) and m = 2*floor((n+1)/2).

a(n) = A342913(A103889(n)). - Wesley Ivan Hurt, May 09 2021

A342913 Pairwise listing of the partitions of 2k into two parts, (s,t), with 0 < t <= s ordered by decreasing values of s and where k = 1,2,... .

Original entry on oeis.org

1, 1, 3, 1, 2, 2, 5, 1, 4, 2, 3, 3, 7, 1, 6, 2, 5, 3, 4, 4, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 11, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6, 13, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 7, 15, 1, 14, 2, 13, 3, 12, 4, 11, 5, 10, 6, 9, 7, 8, 8, 17, 1, 16, 2, 15, 3, 14, 4, 13, 5, 12, 6, 11, 7

Offset: 1

Wesley Ivan Hurt, Mar 28 2021

nonn

			                                                        [13,1]
                                               [11,1]   [12,2]
                                       [9,1]   [10,2]   [11,3]
                               [7,1]   [8,2]   [9, 3]   [10,4]
                       [5,1]   [6,2]   [7,3]   [8, 4]   [9, 5]
               [3,1]   [4,2]   [5,3]   [6,4]   [7, 5]   [8, 6]
       [1,1]   [2,2]   [3,3]   [4,4]   [5,5]   [6, 6]   [7, 7]
   2k    2       4       6       8       10      12       14
  --------------------------------------------------------------------------
   2k   Decreasing partitions of 2k
  --------------------------------------------------------------------------
   2   1,1
   4   3,1,2,2
   6   5,1,4,2,3,3
   8   7,1,6,2,5,3,4,4
  10   9,1,8,2,7,3,6,4,5,5
  12   11,1,10,2,9,3,8,4,7,5,6,6
  14   13,1,12,2,11,3,10,4,9,5,8,6,7,7
  ...

Index entries for sequences related to partitions

Cf. A000194, A052928, A103889, A339399, A339443, A342769.

a(n) = k - (k^2 + k - m)*(-1)^n / 2, where k = round(sqrt(m)) and m = 2*floor((n+1-(-1)^n)/2).

a(n) = A342769(A103889(n)).

A366423 Multiplicative with a(p^e) = p^(e+1-p) if p|e, and p^(e+1) otherwise.

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 16, 27, 100, 121, 18, 169, 196, 225, 8, 289, 108, 361, 50, 441, 484, 529, 144, 125, 676, 3, 98, 841, 900, 961, 64, 1089, 1156, 1225, 54, 1369, 1444, 1521, 400, 1681, 1764, 1849, 242, 675, 2116, 2209, 72, 343, 500, 2601, 338, 2809, 12, 3025

Offset: 1

Amiram Eldar, Nov 17 2023

A permutation of the positive integers. 1 is the only fixed point.

a(n) is a powerful number (A001694) if and only if n is not in A100717.

Cf. A000040, A001694, A005117, A007947, A048102, A072873, A100717, A103889, A130508, A342090.

Similar sequences: A011262, A011264.

f[p_, e_] := p^(e + 1 + If[Mod[e, p] == 0, -p, 0]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

a(2^e) = 2^A103889(e).
a(3^e) = 3^A130508(e).
A007947(a(n)) = A007947(n).
a(A051674(n)) = A000040(n).
a(n) is squarefree (A005117) if and only if n is in A048102.
a(A048102(n)) = A007947(A048102(n)).
a(n) == 0 (mod n) if and only if n is not in A342090.
a(n) | n if and only if n is in A072873.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 1/p + 1/(1 + p) - (p-1)/(p^p * (1 + p^p))) = 0.660264348361... .

cp's OEIS Frontend

A227380 Doubling the first two of every four nonnegative numbers.

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A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

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A307613 Inverse of the permutation A307485: one odd, two even, four odd, eight even, etc; extended with a(0) = 0.

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A121496 Run lengths of successive numbers in A068225.

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A123246 a(n) = PrimePi(n) + (-1)^(PrimePi(n) + 1) (cf. A000720).

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A163975 n-th nonprime - (-1)^(n-th nonprime).

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A226279 a(4n) = a(4n+2) = 2*n , a(4n+1) = a(4n+3) = 2*n-1.

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A226279 a(4n) = a(4n+2) = 2n , a(4n+1) = a(4n+3) = 2n-1.