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 1-3 of 3 results.

A325417 a(n) is the least number not 2*a(m) or 3*a(m)+1 for any m < n.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 19, 20, 21, 23, 27, 29, 31, 32, 33, 35, 36, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 59, 60, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 99, 101, 103, 104
Offset: 1

Views

Author

Clark Kimberling, Apr 24 2019

Keywords

Comments

In column 1 of the following guide to related sequences, disallowed terms are indicated by the variable x representing a(m) for m < n.
Disallowed Sequence(a) Complement(c) Differences(a) Differences(c)
2x, 3x+1 A325417 A325418 A325444 A325445
3x, 2x+1 A325419 A325420 A325494 A325495
2x+1, 3x+1 A077477 A325422 A325496 A325497
2x, 3x A036668 A325424 A325498 A325499
[3x/2], 2x A325425 A325426 A325518 A325519
[3x/2], 2x+1 A325427 A325428 A325520 A325521
[3x/2], 3x A325429 A325430 A325522 A325523
3x, 4x A325431 A325432 A325525 A325526
2x, 3x-1 A325462 A325463 A325526 A325527
2x, 3x-2 A325464 A325465 A325528 A325529
2x-1, 3x-1 A325440 A325441 A325530 A325531
2x-1, 3x A325442 A325443 A325532 A325533
2x+1, 3x+2 A325539 A325540 A325541 A325542

Examples

			The sequence necessarily starts with 1.  The next 2 terms are determined as follows:  because a(1) = 1, the numbers 2 and 4 are disallowed, so that a(2) = 3, whence the numbers 6 and 10 are disallowed, so that a(3) = 5.

Links

Crossrefs

Cf. A325418, A325444.

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 *)

A036668 Hati numbers: of form 2^i*3^j*k, i+j even, (k,6)=1.

Original entry on oeis.org

1, 4, 5, 6, 7, 9, 11, 13, 16, 17, 19, 20, 23, 24, 25, 28, 29, 30, 31, 35, 36, 37, 41, 42, 43, 44, 45, 47, 49, 52, 53, 54, 55, 59, 61, 63, 64, 65, 66, 67, 68, 71, 73, 76, 77, 78, 79, 80, 81, 83, 85, 89, 91, 92, 95, 96, 97, 99, 100, 101, 102, 103, 107
Offset: 1

Views

Author

N. J. A. Sloane, Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)

Keywords

Comments

If n appears then 2n and 3n do not. - Benoit Cloitre, Jun 13 2002
Closed under multiplication. Each term is a product of a unique subset of {6} U A050376 \ {2,3}. - Peter Munn, Sep 14 2019

Links

Crossrefs

Cf. A003159, A007310, A014601, A036667, A050376, A052330, A325424 (complement), A325498 (first differences), A373136 (characteristic function).
Positions of 0's in A182582.
Subsequences: A084087, A339690, A352272, A352273.
Cf. also A121537, A145204, A225837, A307150, A329575, A334747, A334748, A339746.

Programs

  • Maple
    N:= 1000: # to get all terms up to N
A:= {seq(2^i,i=0..ilog2(N))}:
Ae,Ao:= selectremove(issqr,A):
Be:= map(t -> seq(t*9^j, j=0 .. floor(log[9](N/t))),Ae):
Bo:= map(t -> seq(t*3*9^j,j=0..floor(log[9](N/(3*t)))),Ao):
B:= Be union Bo:
C1:= map(t -> seq(t*(6*i+1),i=0..floor((N/t -1)/6)),B):
C2:= map(t -> seq(t*(6*i+5),i=0..floor((N/t - 5)/6)),B):
A036668:= C1 union C2; # Robert Israel, May 09 2014
  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1,
Apply[Or, Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/2}],
IntegerQ]]] &]], {150}]; a  (* A036668 *)
(* Peter J. C. Moses, Apr 23 2019 *)
  • PARI
    twos(n) = {local(r,m);r=0;m=n;while(m%2==0,m=m/2;r++);r}
threes(n) = {local(r,m);r=0;m=n;while(m%3==0,m=m/3;r++);r}
isA036668(n) = (twos(n)+threes(n))%2==0 \\ Michael B. Porter, Mar 16 2010
  • PARI
    is(n)=(valuation(n,2)+valuation(n,3))%2==0 \\ Charles R Greathouse IV, Sep 10 2015
  • PARI
    list(lim)=my(v=List(),N);for(n=0,logint(lim\=1,3),N=if(n%2,2*3^n,3^n); while(N<=lim, forstep(k=N,lim,[4*N,2*N], listput(v,k)); N<<=2)); Set(v) \\ Charles R Greathouse IV, Sep 10 2015
  • Python
    from itertools import count
def A036668(n):
    def f(x):
        c = n+x
        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
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 28 2025

Formula

a(n) = 12/7 * n + O(log^2 n). - Charles R Greathouse IV, Sep 10 2015
{a(n)} = A052330({A014601(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Sep 14 2019

Extensions

Offset changed by Chai Wah Wu, Jan 28 2025

A325499 Difference sequence of A325424.

Original entry on oeis.org

1, 5, 2, 2, 2, 1, 3, 3, 1, 4, 1, 5, 1, 1, 4, 1, 1, 6, 2, 2, 1, 5, 1, 1, 2, 2, 7, 1, 2, 2, 1, 7, 2, 2, 1, 1, 2, 3, 1, 4, 6, 1, 1, 2, 2, 1, 7, 4, 1, 3, 2, 1, 1, 2, 2, 1, 1, 5, 1, 4, 1, 5, 2, 2, 2, 1, 1, 2, 3, 1, 4, 7, 1, 4, 1, 1, 5, 1, 2, 2, 1, 3, 2, 1, 1, 2
Offset: 1

Views

Author

Clark Kimberling, May 05 2019

Keywords

Comments

See A325417 for a guide to related sequences.
Conjecture: every term is in {1,2,3,4,5,6,7}.

Links

Crossrefs

Cf. A325424, A325417, A036668, A325498.

Programs

  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1,
Apply[Or, Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/2}],
IntegerQ]]] &]], {2000}]; a ;       (* A036668 *)
c = Complement[Range[Last[a]], a] ; (* A325424 *)
Differences[a]  (* A325498 *)
Differences[c]  (* A325499 *)
(* Peter J. C. Moses, Apr 23 2019 *)

Formula

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = 12/5. - Amiram Eldar, Nov 26 2020
Showing 1-3 of 3 results.

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

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