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 21-26 of 26 results.

A334840 a(1) = 1, a(n) = a(n-1)/gcd(a(n-1),n) if this gcd is > 1, else a(n) = 4*a(n-1).

Original entry on oeis.org

1, 4, 16, 4, 16, 8, 32, 4, 16, 8, 32, 8, 32, 16, 64, 4, 16, 8, 32, 8, 32, 16, 64, 8, 32, 16, 64, 16, 64, 32, 128, 4, 16, 8, 32, 8, 32, 16, 64, 8, 32, 16, 64, 16, 64, 32, 128, 8, 32, 16, 64, 16, 64, 32, 128, 16, 64, 32, 128, 32, 128, 64, 256, 4, 16
Offset: 1

Views

Author

Ctibor O. Zizka, May 13 2020

Keywords

Comments

A variant of A133058. - Ctibor O. Zizka, Apr 14 2023

Examples

			a(2) = 4*a(1) = 4, a(3) = 4*a(2) = 16, a(4) = a(3)/4 = 4, a(5) = 4*a(4) = 16, ...

Crossrefs

Cf. A000120, A001316, A133058, A255140, A257806, A319018.

Programs

  • Magma
    a:=[1]; for n in [2..70] do if Gcd(a[n-1], n) eq 1 then Append(~a,4* a[n-1]); else Append(~a, a[n-1] div Gcd(a[n-1], n)); end if; end for; a; // Marius A. Burtea, May 13 2020
  • Mathematica
    a[1] = 1; a[n_] := a[n] = If[(g = GCD[a[n-1], n]) > 1, a[n-1]/g, 4*a[n-1]]; Array[a, 100] (* Amiram Eldar, May 13 2020 *)
  • PARI
    lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(g = gcd(va[n-1], n)); if (g > 1, va[n] = va[n-1]/g, va[n] = 4*va[n-1]);); va;} \\ Michel Marcus, May 17 2020
  • Python
    from itertools import count, islice
from math import gcd
def A334840_gen(): # generator of terms
    yield (a:=1)
    for n in count(2):
        yield (a:=a<<2 if (b:=gcd(a,n)) == 1 else a//b)
A334840_list = list(islice(A334840_gen(),30)) # Chai Wah Wu, Mar 18 2023

Formula

a(n) = 2^((n mod 2) + A000120(n) + 1), for n >= 2. - Ctibor O. Zizka, Apr 15 2023
a(n) = 2*A001316(n)*(n mod 2 + 1), for n >= 2. - Ctibor O. Zizka, Apr 15 2023

A334852 a(1) = 1, a(n) = a(n-1) / gcd(a(n-1),n) if this gcd is > 1, else a(n) = a(n-1) + 2.

Original entry on oeis.org

1, 3, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 1, 3, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 1, 3, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49
Offset: 1

Views

Author

Ctibor O. Zizka, May 13 2020

Keywords

Comments

A variant of A133058. For n >= 1, a(n) is an odd number. - Ctibor O. Zizka, Apr 15 2023

Examples

			a(2) = a(1) + 2 = 3, a(3) = a(2)/3 = 1, a(4) = a(3) + 2 = 3, a(5) = a(4) + 2 = 5, ...

Crossrefs

Cf. A133058.

Programs

  • Magma
    a:=[1]; for n in [2..70] do if Gcd(a[n-1], n) eq 1 then Append(~a, a[n-1] + 2); else Append(~a, a[n-1] div Gcd(a[n-1], n)); end if; end for; a; // Marius A. Burtea, May 13 2020
  • Mathematica
    a[1] = 1; a[n_] := a[n] = If[(g = GCD[a[n-1], n]) > 1, a[n-1]/g, a[n-1] + 2]; Array[a, 100] (* Amiram Eldar, May 13 2020 *)
  • PARI
    lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(g = gcd(va[n-1], n)); if (g > 1, va[n] = va[n-1]/g, va[n] = va[n-1]+2);); va;} \\ Michel Marcus, May 17 2020

Formula

From Ctibor O. Zizka, Apr 15 2023: (Start)
For k >= 0:
a(7*2^(2*k + 1) - 13) = 1
a(7*2^(2*k + 1) - 12) = 3
a(7*2^(2*k + 1) - 11) = 1
a(7*2^(2*k + 1) - 10) = 3
a(7*2^(2*k + 1) - 9) = 5
a(7*2^(2*k + 1) - 8) = 7
a(7*2^(2*k + 1) - 7) = 1
a(7*2^(2*k + 1) - 6) = 3
For n from [7*2^(2*k + 1) - 5; 7*2^(2*k + 2) - 10]:
a(n) = 2*t + 1, t from [0; 7*2^(2*k + 1) - 5]
a(7*2^(2*k + 2) - 9) = 1
a(7*2^(2*k + 2) - 8) = 3
For n from [7*2^(2*k + 2) - 7; 7*2^(2*k + 3) - 14]:
a(n) = 2*t + 1, t from [0; 7*2^(2*k + 2) - 7]. (End)

A334942 a(1) = 1, a(n) = a(n-1) / gcd(a(n-1),n) if this gcd is > 1, else a(n) = 2*a(n-1) + 4.

Original entry on oeis.org

1, 6, 2, 1, 6, 1, 6, 3, 1, 6, 16, 4, 12, 6, 2, 1, 6, 1, 6, 3, 1, 6, 16, 2, 8, 4, 12, 3, 10, 1, 6, 3, 1, 6, 16, 4, 12, 6, 2, 1, 6, 1, 6, 3, 1, 6, 16, 1, 6, 3, 1, 6, 16, 8, 20, 5, 14, 7, 18, 3, 10, 5, 14, 7, 18, 3, 10, 5, 14, 1, 6, 1, 6, 3, 1, 6, 16, 8, 20, 1, 6
Offset: 1

Views

Author

Ctibor O. Zizka, May 17 2020

Keywords

Comments

A variant of A133058.

Examples

			a(2) = 2*a(1) + 4 = 6, a(3) = a(2)/3 = 2, a(4) = a(3)/2 = 1, a(5) = 2*a(4) + 4 = 6, ...

Crossrefs

Cf. A133058.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = If[(g = GCD[a[n-1], n]) > 1, a[n-1]/g, 2*a[n-1] + 4]; Array[a, 100]
  • PARI
    lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(g = gcd(va[n-1], n)); if (g > 1, va[n] = va[n-1]/g, va[n] = 2*va[n-1] + 4);); va;} \\ Michel Marcus, May 17 2020

A336164 a(1) = 1; if n>1, and gcd(a(n-1), n) > 1 then a(n) = a(n-1)/gcd(a(n-1), n), otherwise a(n) = a(n-1) + n - 1.

Original entry on oeis.org

1, 2, 4, 1, 5, 10, 16, 2, 10, 1, 11, 22, 34, 17, 31, 46, 62, 31, 49, 68, 88, 4, 26, 13, 37, 62, 88, 22, 50, 5, 35, 66, 2, 1, 35, 70, 106, 53, 91, 130, 170, 85, 127, 170, 34, 17, 63, 21, 3, 52, 102, 51, 103, 156, 210, 15, 5, 62, 120, 2, 62, 1, 63, 126, 190, 95, 161, 228, 76
Offset: 1

Views

Author

Todor Szimeonov, Jul 10 2020

Keywords

Crossrefs

Cf. A133058.

Programs

  • PARI
    a(n) = if (n==1, 1, my(prec=a(n-1)); if (gcd(prec, n) > 1, prec/gcd(prec,n), n-1+prec)); \\ Michel Marcus, Jul 13 2020
  • Python
    from itertools import count, islice
from math import gcd
def A336164_gen(): # generator of terms
    yield (a:=1)
    for n in count(2):
        yield (a:=a+n-1 if (b:=gcd(a,n)) == 1 else a//b)
A336164_list = list(islice(A336164_gen(),30)) # Chai Wah Wu, Mar 18 2023

A361672 a(1) = a(2) = 1; for n > 2, a(n) = a(n-2) + a(n-1) + n if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1), n).

Original entry on oeis.org

1, 1, 5, 10, 2, 1, 10, 5, 24, 12, 47, 71, 131, 216, 72, 9, 98, 49, 166, 83, 270, 135, 428, 107, 560, 280, 867, 1175, 2071, 3276, 5378, 2689, 8100, 4050, 810, 45, 892, 446, 1377, 1863, 3281, 5186, 8510, 4255, 851, 37, 935, 1020, 2004, 1002, 334, 167, 554, 277, 886
Offset: 1

Views

Author

Hubert W. Westwood, Mar 20 2023

Keywords

Crossrefs

Cf. A133058, A091508, A133579, A133580, A255051, A255140.

Programs

  • Magma
    a:=[1, 1]; for n in [3..50] do if Gcd(a[n-1], n) eq 1 then Append(~a, a[n-2] + a[n-1] + n); else Append(~a, a[n-1] div Gcd(a[n-1], n)); end if; end for; [] cat a;
  • Mathematica
    a[1]=a[2]=1; a[n_]:=a[n]=If[GCD[a[n-1],n]==1,a[n-2]+a[n-1]+n,a[n-1]/GCD[a[n-1],n]]; Array[a,55] (* Stefano Spezia, Mar 20 2023 *)

A369543 a(0) = 1; for n >= 0, a(n+1) = n - a(n) if a(n) odd, else a(n+1) = floor((3*n + a(n))/2).

Original entry on oeis.org

1, -1, 2, 4, 6, 9, -4, 7, 0, 12, 19, -9, 20, 28, 33, -19, 34, 41, -24, 15, 4, 32, 47, -25, 48, 60, 67, -41, 68, 76, 81, -51, 82, 89, -56, 23, 12, 60, 85, -47, 86, 103, -62, 32, 80, 106, 120, 129, -82, 31, 18, 84, 118, 137, -84, 39, 16, 92, 131, -73, 132, 156, 169, -107
Offset: 0

Views

Author

Ctibor O. Zizka, Jan 25 2024

Keywords

Comments

After a chaotic part, at n = 358 the sequence settles down and becomes quasi-periodic with a 6-loop. For some choices of the initial term a(0) the sequence stays chaotic.

Examples

			For n = 0, a(0) = 1.
For n = 1, a(0) is odd, thus a(1) = 0 - 1 = -1.
For n = 2, a(1) is odd, thus a(2) = 1 - (-1) = 2.
For n = 3, a(2) is even, thus a(3) = floor((3*2 + a(2))/2) = 4.
etc.

Links

Crossrefs

Cf. A133058.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = If[OddQ[a[n - 1]], n - 1 - a[n - 1], Floor[(3*n - 3 + a[n - 1])/2]]; Array[a, 100, 0] (* Amiram Eldar, Jan 26 2024 *)

Formula

For k >= 0 :
a(358 + 6*k) = 1062 + 18*k.
a(359 + 6*k) = 1068 + 18*k.
a(360 + 6*k) = 1072 + 18*k.
a(361 + 6*k) = 1076 + 18*k.
a(362 + 6*k) = 1079 + 18*k.
a(363 + 6*k) = -717 - 12*k.
Previous Showing 21-26 of 26 results.

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