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 1-2 of 2 results.

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

A382748 Primitive exponents for the greedy convolution of length 4.

Original entry on oeis.org

1, 5, 6, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 30, 31, 32, 35, 36, 37, 40, 41, 42, 43, 45, 47, 48, 49, 53, 55, 56, 59, 61, 63, 65, 66, 67, 71, 73, 77, 78, 79, 83, 85, 88, 89, 91, 95, 97, 99, 101, 102, 103, 104, 107, 109, 113, 114, 115, 117, 119, 121, 125, 127, 131, 133, 135, 136, 137, 138, 139
Offset: 1

Views

Author

Jan Snellman, Apr 24 2025

Keywords

Comments

Integers n such that p^n is primitive for the "greedy convolution of length 4" when p is a prime number.
Smallest elements in the blocks of the greedy partition of the positive integers into parts of length at most 4.
First column of A382747.
Strict definition:
Form an infinite rooted tree T on the nonnegative integers in the following way.
1. 0 is the root
2. Form a branch 0 - 1 - 2 -3 - 4
3. Proceed inductively. Add n to end of an existing branch as either
0 - k=n
0 - k - 2k=n
0 - k - 2k - 3k=n
0 - k - 2k - 3k - 4k=n
with a preference for smaller k.
The primitive elements are the integers at distance one from the root.

Examples

			Up to n=15 the branches of the aforementioned tree looks like
  0 - 1 - 2 - 3 - 4
  0 - 5 - 10 - 15
  0 - 6 - 12
  0 - 7 - 14
  0 - 8
  0 - 9
  0 - 11
  0 - 13
so the primitive elements <= 15 are 1, 5, 6, 7, 8, 9, 11, 13.

Links

Crossrefs

First column of A382747.
Cf. A121537.

Programs

Showing 1-2 of 2 results.

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

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