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 1-5 of 5 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

A025620 Numbers of the form 4^i * 9^j, with i, j >= 0.

Original entry on oeis.org

1, 4, 9, 16, 36, 64, 81, 144, 256, 324, 576, 729, 1024, 1296, 2304, 2916, 4096, 5184, 6561, 9216, 11664, 16384, 20736, 26244, 36864, 46656, 59049, 65536, 82944, 104976, 147456, 186624, 236196, 262144, 331776, 419904, 531441, 589824, 746496, 944784
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Numbers of the form 2^(2*i) * 3^(2*j) or 3-smooth squares: intersection of A003586 and A000290; A001221(a(n)) <= 2; A001222(a(n)) is even; A006530(a(n)) <= 3. - Reinhard Zumkeller, May 16 2015
Closed under multiplication. - Klaus Purath, Oct 06 2023

Links

Crossrefs

Cf. A003586, A000290, A001221, A001222, A006530, subsequence of A036667.
Cf. A022328, A022329.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a025620 n = a025620_list !! (n-1)
a025620_list = f $ singleton 1 where
   f s = y : f (insert (4 * y) $ insert (9 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015
  • Mathematica
    n = 10^6; Flatten[Table[4^i*9^j, {i, 0, Log[4, n]}, {j, 0, Log[9, n/4^i]}]] // Sort (* Amiram Eldar, Sep 24 2020 *)
  • PARI
    list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 9), N=9^n; while(N<=lim, listput(v, N); N<<=2)); Set(v) \\ Charles R Greathouse IV, Jan 10 2018

Formula

Sum_{n>=1} 1/a(n) = (4*9)/((4-1)*(9-1)) = 3/2. - Amiram Eldar, Sep 24 2020
a(n) ~ exp(sqrt(8*log(2)*log(3)*n)) / 6 . - Vaclav Kotesovec, Sep 24 2020
a(n) = A003586(n)^2 = 4^A022328(n) * 9^A022329(n). - R. J. Mathar, Jul 06 2025

A257999 Numbers of the form, 2^i*3^j, i+j odd.

Original entry on oeis.org

2, 3, 8, 12, 18, 27, 32, 48, 72, 108, 128, 162, 192, 243, 288, 432, 512, 648, 768, 972, 1152, 1458, 1728, 2048, 2187, 2592, 3072, 3888, 4608, 5832, 6912, 8192, 8748, 10368, 12288, 13122, 15552, 18432, 19683, 23328, 27648, 32768, 34992, 41472, 49152, 52488
Offset: 1

Views

Author

Reinhard Zumkeller, May 16 2015

Keywords

Links

Crossrefs

Complement of A036667 with respect to A003586.
Intersection of A026424 and A003586.
Cf. A069352, A022328, A022329.

Programs

  • Haskell
    a257999 n = a257999_list !! (n-1)
a257999_list = filter (odd . flip mod 2 . a001222) a003586_list
  • Mathematica
    max = 53000; Reap[Do[k = 2^i*3^j; If[k <= max && OddQ[i + j], Sow[k]], {i, 0, Log[2, max] // Ceiling}, {j, 0, Log[3, max] // Ceiling}]][[2, 1]] // Union (* Amiram Eldar, Feb 18 2021 after Jean-François Alcover at A036667 *)
  • Python
    from sympy import integer_log
def A257999(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): return n+x-sum((x//3**i).bit_length()+(i&1)>>1 for i in range(integer_log(x, 3)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

Formula

A069352(a(n)) mod 2 = 1.
Sum_{n>=1} 1/a(n) = 5/4. - Amiram Eldar, Feb 18 2021

A054774 Numbers of the form 2^i*5^j where i+j is odd.

Original entry on oeis.org

2, 5, 8, 20, 32, 50, 80, 125, 128, 200, 320, 500, 512, 800, 1250, 1280, 2000, 2048, 3125, 3200, 5000, 5120, 8000, 8192, 12500, 12800, 20000, 20480, 31250, 32000, 32768, 50000, 51200, 78125, 80000, 81920, 125000, 128000, 131072, 200000
Offset: 1

Views

Author

Henry Bottomley, May 19 2000

Keywords

Comments

m is in the sequence iff m/10 is already in the sequence or m is an odd power of 2 or 5.

Links

Crossrefs

Intersection of A026424 and A003592.
Cf. A004171, A013710.

Programs

  • Mathematica
    max = 200000; Reap[Do[k = 2^i*5^j; If[k <= max && OddQ[i + j], Sow[k]], {i, 0, Log[2, max] // Ceiling}, {j, 0, Log[5, max] // Ceiling}]][[2, 1]] // Union (* Amiram Eldar, Feb 18 2021 after Jean-François Alcover at A036667 *)

Formula

Sum_{n>=1} 1/a(n) = 35/36. - Amiram Eldar, Feb 18 2021

A054901 Numbers of the form 2^i*5^j where i+j is even.

Original entry on oeis.org

1, 4, 10, 16, 25, 40, 64, 100, 160, 250, 256, 400, 625, 640, 1000, 1024, 1600, 2500, 2560, 4000, 4096, 6250, 6400, 10000, 10240, 15625, 16000, 16384, 25000, 25600, 40000, 40960, 62500, 64000, 65536, 100000, 102400, 156250, 160000, 163840
Offset: 1

Views

Author

Henry Bottomley, May 23 2000

Keywords

Comments

m is in the sequence iff m/10 is already in the sequence or m is a power of 4 or 25.

Links

Crossrefs

Intersection of A028260 and A003592.
Cf. A000302, A009969, A054774.

Programs

  • Mathematica
    max = 200000; Reap[Do[k = 2^i*5^j; If[k <= max && EvenQ[i + j], Sow[k]], {i, 0, Log[2, max] // Ceiling}, {j, 0, Log[5, max] // Ceiling}]][[2, 1]] // Union (* Amiram Eldar, Feb 18 2021 after Jean-François Alcover at A036667 *)

Formula

Sum_{n>=1} 1/a(n) = 55/36. - Amiram Eldar, Feb 18 2021
Showing 1-5 of 5 results.

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

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