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 1-10 of 50 results. Next

A325418 Complement of A325417.

Original entry on oeis.org

2, 4, 6, 10, 14, 16, 18, 22, 24, 25, 26, 28, 30, 34, 37, 38, 40, 42, 46, 52, 54, 58, 61, 62, 64, 66, 70, 72, 78, 82, 86, 88, 90, 94, 96, 97, 98, 100, 102, 106, 109, 110, 112, 114, 118, 120, 124, 126, 130, 133, 134, 136, 138, 142, 145, 146, 148, 150, 151, 152
Offset: 1

Views

Author

Clark Kimberling, Apr 24 2019

Keywords

References

  • These are the numbers 2x and 3x+1 as x ranges through the numbers in A325417.

Links

Crossrefs

Cf. A325417.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1,
Apply[Or, Map[MemberQ[a, #] &, Select[Flatten[{#/2, (# - 1)/3}],
IntegerQ]]] &]], {150}]; a     (* A325417 *)
Complement[Range[Last[a]], a]  (* A325418 *)
(* Peter J. C. Moses, Apr 23 2019 *)

A325444 Difference sequence of A325417.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 03 2019

Keywords

Comments

See A325417 for a guide to related sequences. Conjecture: all the differences are in {1,2,3,4}; a count of differences d(n) = a(n)-a(n-1) for n=2..10000 follows: 4755 occurrences of d(n) = 1; 4332 of 2; 744 of 3; and 169 of 4.

Examples

			A325417 is given by A(n) = least number not 2*A(m) or 3*A(m)+1 for any m < n, so that A = (1,3,5,7,8,9,11, ...), with differences (2,2,2,1,1,2,...).

Links

Crossrefs

Cf. A325417, A325445.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1,
Apply[Or, Map[MemberQ[a, #] &, Select[Flatten[{#/2, (# - 1)/3}],
IntegerQ]]] &]], {2000}]; a ;       (* A325417 *)
c = Complement[Range[Last[a]], a] ; (* A325418 *)
Differences[a]  (* A325444 *)
Differences[c]  (* A325445 *)
(* Peter J. C. Moses, Apr 23 2019 *)

A325424 Complement of A036668: numbers not of the form 2^i*3^j*k, i + j even, (k,6) = 1.

Original entry on oeis.org

2, 3, 8, 10, 12, 14, 15, 18, 21, 22, 26, 27, 32, 33, 34, 38, 39, 40, 46, 48, 50, 51, 56, 57, 58, 60, 62, 69, 70, 72, 74, 75, 82, 84, 86, 87, 88, 90, 93, 94, 98, 104, 105, 106, 108, 110, 111, 118, 122, 123, 126, 128, 129, 130, 132, 134, 135, 136, 141, 142
Offset: 1

Views

Author

Clark Kimberling, Apr 26 2019

Keywords

Comments

These are the numbers 2x and 3x as x ranges through the numbers in A036668.
Numbers whose squarefree part is divisible by exactly one of {2, 3}. - Peter Munn, Aug 24 2020
The asymptotic density of this sequence is 5/12. - Amiram Eldar, Sep 20 2020

Links

Crossrefs

Cf. A325417, A036668.
Symmetric difference of: A003159 and A007417; A036554 and A145204\{0}.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/2}],
IntegerQ]]] &]], {150}]; a     (* A036668 *)
Complement[Range[Last[a]], a]  (* A325424 *)
(* Peter J. C. Moses, Apr 23 2019 *)
  • Python
    from itertools import count
def A325424(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c += (m-1)//6+(m-5)//6+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 28 2025

Formula

(2 * {A036668}) union (3 * {A036668}). - Sean A. Irvine, May 19 2019

A325432 Complement of A325431.

Original entry on oeis.org

3, 4, 6, 8, 15, 20, 21, 27, 28, 30, 33, 36, 39, 40, 42, 44, 48, 51, 52, 54, 56, 57, 64, 66, 68, 69, 72, 75, 76, 78, 87, 88, 92, 93, 96, 100, 102, 104, 105, 111, 114, 116, 123, 124, 128, 129, 135, 136, 138, 140, 141, 147, 148, 150, 152, 159, 164, 165, 172
Offset: 1

Views

Author

Clark Kimberling, May 01 2019

Keywords

Comments

These are the numbers 3x and 4x as x ranges through the numbers x > 1 in A325431.
Equivalently, numbers k whose exponent of the highest power of 3 dividing k and exponent of the highest power of 4 dividing k have an opposite parity. The asymptotic density of this sequence is 7/20. - Amiram Eldar, Sep 20 2020

Links

Crossrefs

Cf. A325417, A325431.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/4}],
IntegerQ]]] &]], {150}]; a          (* A325431 *)
Complement[Range[Last[a]], a]       (* A325432 *)
(* Peter J. C. Moses, Apr 25 2019 *)
Select[Range[100], !Equal @@ Mod[IntegerExponent[#, {3, 4}], 2] &] (* Amiram Eldar, Sep 20 2020 *)

A325431 a(n) is the least number not 3*a(m) or 4*a(m) for any m < n.

Original entry on oeis.org

1, 2, 5, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 29, 31, 32, 34, 35, 37, 38, 41, 43, 45, 46, 47, 49, 50, 53, 55, 58, 59, 60, 61, 62, 63, 65, 67, 70, 71, 73, 74, 77, 79, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91, 94, 95, 97, 98, 99, 101
Offset: 1

Views

Author

Clark Kimberling, May 01 2019

Keywords

Comments

Equivalently, numbers k whose exponent of the highest power of 3 dividing k and exponent of the highest power of 4 dividing k have the same parity. The asymptotic density of this sequence is 13/20. - Amiram Eldar, Sep 20 2020

Examples

			The sequence necessarily starts with 1.  The next 2 terms are determined as follows:  because a(1) = 1, the numbers 3 and 4 are disallowed, so that a(2) = 2, whence the numbers 6 and 8 are disallowed, and a(3) = 5.  See A325417 for a guide to related sequences.

Links

Crossrefs

Cf. A325417, A325432.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/4}],
IntegerQ]]] &]], {150}]; a          (* A325431 *)
Complement[Range[Last[a]], a]       (* A325432 *)
(* Peter J. C. Moses, Apr 25 2019 *)
Select[Range[100], Equal @@ Mod[IntegerExponent[#, {3, 4}], 2] &] (* Amiram Eldar, Sep 20 2020 *)

A325426 Complement of A325425.

Original entry on oeis.org

2, 4, 6, 7, 10, 12, 13, 16, 18, 21, 22, 25, 28, 30, 34, 36, 38, 39, 40, 43, 46, 48, 49, 52, 54, 55, 58, 61, 62, 63, 64, 66, 67, 70, 74, 75, 76, 79, 82, 84, 85, 88, 90, 94, 97, 100, 102, 103, 106, 108, 109, 112, 114, 115, 117, 118, 120, 121, 124, 129, 130
Offset: 1

Views

Author

Clark Kimberling, Apr 27 2019

Keywords

Comments

These are the numbers 2x and floor(3x/2) as x ranges through the numbers x > 1 in A325425.

Links

Crossrefs

Cf. A325417, A325425.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/2,
If[Mod[#, 3] == 0, (2 #)/3, 0] + If[Mod[#, 3] == 1, 1/3 (1 + 2 #), 0]}],
IntegerQ || # == 0]]] &]], {150}]; a (* A325425 *)
Complement[Range[Last[a]], a];       (* A325426 *)
(* - Peter J. C. Moses, Apr 25 2019 *)

A325419 a(n) is the least number not 2*a(m)+1 or 3*a(m) for any m < n.

Original entry on oeis.org

1, 2, 4, 7, 8, 10, 11, 13, 14, 16, 18, 19, 20, 22, 25, 26, 28, 31, 32, 34, 35, 36, 38, 40, 43, 44, 46, 47, 49, 50, 52, 55, 56, 58, 59, 61, 62, 64, 67, 68, 70, 72, 74, 76, 79, 80, 82, 83, 85, 86, 88, 90, 91, 92, 94, 97, 98, 100, 103, 104, 106, 107, 109, 110
Offset: 1

Views

Author

Clark Kimberling, Apr 28 2019

Keywords

Examples

			The sequence necessarily starts with 1.  The next 2 terms are determined as follows:  because a(1) = 1, the number 3 is disallowed, so that a(2) = 2, whence the numbers 5 and 6 are disallowed, and a(3) = 4.  See A325417 for a guide to related sequences.

Links

Crossrefs

Cf. A325420.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/3, (# - 1)/2}],
IntegerQ]]] &]], {150}]; a;     (* A325419 *)
Complement[Range[Last[a]], a];  (* A325420 *)
(* Peter J. C. Moses, Apr 23 2019 *)

A325420 Complement of A325419.

Original entry on oeis.org

3, 5, 6, 9, 12, 15, 17, 21, 23, 24, 27, 29, 30, 33, 37, 39, 41, 42, 45, 48, 51, 53, 54, 57, 60, 63, 65, 66, 69, 71, 73, 75, 77, 78, 81, 84, 87, 89, 93, 95, 96, 99, 101, 102, 105, 108, 111, 113, 114, 117, 119, 120, 123, 125, 129, 132, 135, 137, 138, 141, 145
Offset: 1

Views

Author

Clark Kimberling, Apr 28 2019

Keywords

Comments

These are the numbers 2x+1 and 3x as x ranges through the numbers in A325419.

Links

Crossrefs

Cf. A325417, A325419.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/3, (# - 1)/2}],
IntegerQ]]] &]], {150}]; a;     (* A325419 *)
Complement[Range[Last[a]], a];  (* A325420 *)
(* Peter J. C. Moses, Apr 23 2019 *)

A325425 a(n) is the least number not 2*a(m) or floor(3*a(m)/2) for any m < n.

Original entry on oeis.org

1, 3, 5, 8, 9, 11, 14, 15, 17, 19, 20, 23, 24, 26, 27, 29, 31, 32, 33, 35, 37, 41, 42, 44, 45, 47, 50, 51, 53, 56, 57, 59, 60, 65, 68, 69, 71, 72, 73, 77, 78, 80, 81, 83, 86, 87, 89, 91, 92, 93, 95, 96, 98, 99, 101, 104, 105, 107, 110, 111, 113, 116, 119
Offset: 1

Views

Author

Clark Kimberling, Apr 27 2019

Keywords

Comments

The sequence necessarily starts with 1. The next 2 terms are determined as follows: because a(1) = 1, the number 2 is disallowed, so that a(2) = 3, whence the numbers 4 and 6 are disallowed, and a(3) = 5. See A325417 for a guide to related sequences.

Links

Crossrefs

Cf. A325417, A325426.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/2,
If[Mod[#, 3] == 0, (2 #)/3, 0] + If[Mod[#, 3] == 1, 1/3 (1 + 2 #), 0]}],
IntegerQ || # == 0]]] &]], {150}]; a (* A325425 *)
Complement[Range[Last[a]], a];       (* A325426 *)
(* Peter J. C. Moses, Apr 25 2019 *)

A325427 a(n) is the least number not 2*a(m)+1 or floor(3*a(m)/2) for any m < n.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 18, 20, 22, 24, 25, 26, 28, 31, 32, 34, 35, 38, 40, 43, 44, 47, 50, 54, 55, 56, 58, 59, 61, 62, 67, 68, 72, 73, 74, 76, 78, 79, 80, 83, 85, 86, 90, 92, 94, 96, 97, 98, 99, 103, 104, 105, 106, 107, 110, 112, 115, 116, 121, 122, 126
Offset: 1

Views

Author

Clark Kimberling, Apr 30 2019

Keywords

Examples

			The sequence necessarily starts with 1.  The next 2 terms are determined as follows:  because a(1) = 1, the number 3 is disallowed, so that a(2) = 2, whence the numbers 5 and 6 are disallowed, and a(3) = 4.  See A325417 for a guide to related sequences.

Links

Crossrefs

Cf. A325417, A325428.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{(#-1)/2,
If[Mod[#, 3] == 0, (2 #)/3, 0] + If[Mod[#, 3] == 1, 1/3 (1 + 2 #), 0]}],
IntegerQ || # == 0]]] &]], {150}]; a  (* A325427 *)
Complement[Range[Last[a]], a]         (* A325428 *)
(* Peter J. C. Moses, Apr 25 2019 *)
Showing 1-10 of 50 results. Next

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