A225837 -id:A225837 - cp's OEIS frontend

A036668 Hati numbers: of form 2^i3^jk, i+j even, (k,6)=1.

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

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

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

Robert Israel, Table of n, a(n) for n = 0..10000
Don McDonald, Obituary of Alan Robert Boyd, posted to sci.math Jan 02 1999; alternative link.

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.

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

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 *)

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

is(n)=(valuation(n,2)+valuation(n,3))%2==0 \\ Charles R Greathouse IV, Sep 10 2015

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

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

A225838 Numbers of form 2^i3^j(6k+5), i, j, k >= 0.

Original entry on oeis.org

5, 10, 11, 15, 17, 20, 22, 23, 29, 30, 33, 34, 35, 40, 41, 44, 45, 46, 47, 51, 53, 58, 59, 60, 65, 66, 68, 69, 70, 71, 77, 80, 82, 83, 87, 88, 89, 90, 92, 94, 95, 99, 101, 102, 105, 106, 107, 113, 116, 118, 119, 120, 123, 125, 130, 131, 132, 135, 136, 137, 138

Offset: 1

Ralf Stephan, May 16 2013

Are a(n) > A225837(n) for all n? - Zak Seidov, May 17 2013
Yes. Imagine every 3-smooth number, m, visits you regularly, depositing a gold coin for safe keeping at each epoch (6k+1)*m and collecting it at epoch (6k+5)*m. If you run out of coins, you are doing something other than keeping them in a vault! - Peter Munn, Nov 13 2023
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022

Zak Seidov, Table of n, a(n) for n = 1..10000
Zak Seidov, Graph of A225838(n) - A225837(n) for n=1..126454.

Complement of A225837.

Symmetric difference of A003159 and A026225; of A189716 and A325424.

[n: n in [1..200] | d mod 6 eq 5 where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // Bruno Berselli, May 16 2013

mx = 153; t = {}; Do[n = 2^i*3^j (6 k + 5); If[n <= mx, AppendTo[t, n]], {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, mx/6}]; Union[t] (* T. D. Noe, May 16 2013 *)

for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%6==5),print1(n,",")))

from sympy import integer_log
def A225838(n):
    def f(x): return n+sum(((x//3**i>>j)+5)//6 for i in range(integer_log(x,3)[0]+1) for j in range((x//3**i).bit_length()))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 02 2025

A352272 Numbers whose squarefree part is congruent to 1 modulo 6.

Original entry on oeis.org

1, 4, 7, 9, 13, 16, 19, 25, 28, 31, 36, 37, 43, 49, 52, 55, 61, 63, 64, 67, 73, 76, 79, 81, 85, 91, 97, 100, 103, 109, 112, 115, 117, 121, 124, 127, 133, 139, 144, 145, 148, 151, 157, 163, 169, 171, 172, 175, 181, 187, 193, 196, 199, 205, 208, 211, 217, 220, 223, 225, 229

Offset: 1

Peter Munn, Mar 10 2022

Numbers of the form 4^i * 9^j * (6k+1), i, j, k >= 0.
Closed under multiplication.
The sequence forms a subgroup of the positive integers under the commutative operation A059897(.,.), one of 8 subgroups of the form {k : A007913(k) == 1 (mod m)} - in each case m is a divisor of 24. A059897 has a relevance to squarefree parts that arises from the identity A007913(k*n) = A059897(A007913(k), A007913(n)), where A007913(n) is the squarefree part of n.
The subgroup has 8 cosets, which partition the positive integers as follows. For each i in {1, 5}, j in {1, 2, 3, 6} there is a coset {m^2 * (6k+i) * j : m >= 1, k >= 0}. See the table in the examples.
None of the 8 cosets have been entered into the database previously, but many subgroups of the quotient group (which are formed of certain combinations of cosets) are represented among earlier OEIS sequences, including 6 of the 7 subgroups of index 2 (which combine 4 cosets). This sequence can therefore be defined as the intersection of pairs or triples of these sequences in many combinations (see the cross-references). See also the table in the example section of A352273 (the coset that includes 5).

			The squarefree part of 9 is 1, which is congruent to 1 (mod 6), so 9 is in the sequence.
The squarefree part of 14 is 14, which is congruent to 2 (mod 6), so 14 is not in the sequence.
The squarefree part of 52 = 2^2 * 13 is 13, which is congruent to 1 (mod 6), so 52 is in the sequence.
The 8 cosets described in the initial comments (forming a partition of the positive integers) are shown as rows of the following table. The first half of the table corresponds to (6k+i) with i=1; the second half to i=5, with row 5 being A352273.
   1,  4,   7,   9,  13,  16,  19,  25,  28,  31,  36, ...
   2,  8,  14,  18,  26,  32,  38,  50,  56,  62,  72, ...
   3, 12,  21,  27,  39,  48,  57,  75,  84,  93, 108, ...
   6, 24,  42,  54,  78,  96, 114, 150, 168, 186, 216, ...
   5, 11,  17,  20,  23,  29,  35,  41,  44,  45,  47, ...
  10, 22,  34,  40,  46,  58,  70,  82,  88,  90,  94, ...
  15, 33,  51,  60,  69,  87, 105, 123, 132, 135, 141, ...
  30, 66, 102, 120, 138, 174, 210, 246, 264, 270, 282, ...
The product of two positive integers is in this sequence if and only if they are in the same coset. The asymptotic density of cosets (containing) 1 and 5 is 1/4; of cosets 2 and 10 is 1/8; of cosets 3 and 15 is 1/12; of cosets 6 and 30 is 1/24.

Eric Weisstein's World of Mathematics, Group, Quotient Group, Squarefree Part.

Intersection of any 2 of A055047, A339690 and A352274.
Intersection of any 4 sets chosen from A003159, A007417, A026225, A036668, A189715 and A225837 (in most cases, only 3 sets are needed - specifically if the pairwise intersections of the 3 sets differ from each other).
Closure of A084089 under multiplication by 9.
Other subsequences: A000290\{0}, A016921, A229848 (apparently, with 55 the first difference).
A334832 lists equivalent sequences modulo other divisors of 24.
Cf. A007913, A059897, A307151, A352273.

PARI
```
isok(m) = core(m) % 6 == 1;
```

from itertools import count
def A352272(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+x
        for i in count(0):
            i2 = 9**i
            if i2>x:
                break
            for j in count(0,2):
                k = i2<x:
                    break
                c -= (x//k-1)//6+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

{a(n) : n >= 1} = {m >= 1 : A007913(m) == 1 (mod 6)}.
{a(n) : n >= 1} = A334832 U A334832/7 U A334832/13 U A334832/19 where A334832/k denotes {A334832(m)/k : m >= 1, k divides A334832(m)}.
Using the same denotation, {a(n) : n >= 1} = A352273/5 = {A307151(A352273(m)) : m >= 1}.

A352273 Numbers whose squarefree part is congruent to 5 modulo 6.

Original entry on oeis.org

5, 11, 17, 20, 23, 29, 35, 41, 44, 45, 47, 53, 59, 65, 68, 71, 77, 80, 83, 89, 92, 95, 99, 101, 107, 113, 116, 119, 125, 131, 137, 140, 143, 149, 153, 155, 161, 164, 167, 173, 176, 179, 180, 185, 188, 191, 197, 203, 207, 209, 212, 215, 221, 227, 233, 236, 239, 245, 251

Offset: 1

Peter Munn, Mar 10 2022

Numbers of the form 4^i * 9^j * (6k+5), i, j, k >= 0.
1/5 of each multiple of 5 in A352272.
The product of any two terms is in A352272.
The product of a term of this sequence and a term of A352272 is a term of this sequence.
The positive integers are usefully partitioned as {A352272, 2*A352272, 3*A352272, 6*A352272, {a(n)}, 2*{a(n)}, 3*{a(n)}, 6*{a(n)}}. There is a table in the example section giving sequences formed from unions of the parts.
The parts correspond to the cosets of A352272 considered as a subgroup of the positive integers under the operation A059897(.,.). Viewed another way, the parts correspond to the intersection of the integers with the cosets of the multiplicative subgroup of the positive rationals generated by the terms of A352272.
The asymptotic density of this sequence is 1/4. - Amiram Eldar, Apr 03 2022

			The squarefree part of 11 is 11, which is congruent to 5 (mod 6), so 11 is in the sequence.
The squarefree part of 15 is 15, which is congruent to 3 (mod 6), so 15 is not in the sequence.
The squarefree part of 20 = 2^2 * 5 is 5, which is congruent to 5 (mod 6), so 20 is in the sequence.
The table below lists OEIS sequences that are unions of the cosets described in the initial comments, and indicates the cosets included in each sequence. A352272 (as a subgroup) is denoted H, and this sequence (as a coset) is denoted H/5, in view of its terms being one fifth of the multiples of 5 in A352272.
             H    2H    3H    6H    H/5  2H/5  3H/5  6H/5
A003159      X           X           X           X
A036554            X           X           X           X
.
A007417      X     X                 X     X
A145204\{0}              X     X                 X     X
.
A026225      X           X                 X           X
A026179\{1}        X           X     X           X
.
A036668      X                 X     X                 X
A325424            X     X                 X     X
.
A055047      X                             X
A055048            X                 X
A055041                  X                             X
A055040                        X                 X
.
A189715      X                 X           X     X
A189716            X     X           X                 X
.
A225837      X     X     X     X
A225838                              X     X     X     X
.
A339690      X                       X
A329575                  X                       X
.
A352274      X           X
(The sequence groupings in the table start with the subgroup of the quotient group of H, followed by its cosets.)

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree Part.

Intersection of any three of A003159, A007417, A189716 and A225838.
Intersection of A036668 and A055048.
Complement within A339690 of A352272.
Cf. A007913, A059897, A307151, A334832.
Closure of A084088 under multiplication by 9.
Other subsequences: A033429\{0}, A016969.
Other sequences in the example table: A036554, A145204, A026179, A026225, A325424, A055040, A055041, A055047, A189715, A225837, A329575, A352274.

q[n_] := Module[{e2, e3}, {e2, e3} = IntegerExponent[n, {2, 3}]; EvenQ[e2] && EvenQ[e3] && Mod[n/2^e2/3^e3, 6] == 5]; Select[Range[250], q] (* Amiram Eldar, Apr 03 2022 *)

PARI
```
isok(m) = core(m) % 6 == 5;
```

from itertools import count
def A352273(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+x
        for i in count(0):
            i2 = 9**i
            if i2>x: break
            for j in count(0,2):
                k = i2<x: break
                c -= (x//k-5)//6+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

{a(n) : n >= 1} = {m >= 1 : A007913(m) == 5 (mod 6)}.
{a(n) : n >= 1} = A334832/5 U A334832/11 U A334832/17 U A334832/23 where A334832/k denotes {A334832(m)/k : m >= 1, k divides A334832(m)}.
Using the same notation, {a(n) : n >= 1} = A352272/5 = {A307151(A352272(m)) : m >= 1}.
{A225838(n) : n >= 1} = {m : m = a(j)*k, j >= 1, k divides 6}.

A352274 Numbers whose squarefree part is congruent to 1 modulo 6 or 3 modulo 18.

Original entry on oeis.org

1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 52, 55, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 85, 91, 93, 97, 100, 103, 108, 109, 111, 112, 115, 117, 121, 124, 127, 129, 133, 139, 144, 145, 147, 148, 151, 156, 157, 163, 165, 169, 171, 172

Offset: 1

Peter Munn, Mar 10 2022

Numbers of the form 4^i * 3^j * (6k+1), i, j, k >= 0. Numbers whose prime factorization has an even number of factors of 2 and an even number of factors of the form 6k+5 (therefore also an even number of factors of the form 3k+2).
Closed under multiplication.
Includes the nonzero Loeschian numbers (A003136). The two sequences have few early differences (the first extra number here is a(22) = 55, followed by 85, 115, 145, ...), but their densities diverge progressively, driven by the presence here - and absence from A003136 - of the nonsquare terms of A108166. Asymptotic densities are 1/3 and 0 respectively.
Term by term, the sequence is one half of its complement within A225837.

			4 = 2^2 has square part 2^2, therefore squarefree part 4/2^2 = 1, which is congruent to 1 mod 6, so 4 is in the sequence.
63 = 3^2 * 7 has square part 3^2, therefore squarefree part 63/3^2 = 7, which is congruent to 1 mod 6, so 63 is in the sequence.
21 = 3*7 has square part 1^2 and squarefree part 21, which is congruent to 3 mod 18, so 21 is in the sequence.
72 = 2^3 * 3^2 has square part 2^2 * 3^2 = 6^2, therefore squarefree part 72/6^2 = 2, which is congruent to 2 mod 6 and to 2 mod 18, so 72 is not in the sequence.

Eric Weisstein's World of Mathematics, Squarefree Part.

Intersection of any two of A003159, A026225 and A225837.
Closure of A084089 under multiplication by 3.
Cf. A007913.
Subsequences: A003136\{0}, A108166, A352272.

isok(m) = core(m) % 6 == 1 || core(m) % 18 == 3;

from itertools import count
def A352274(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+x
        for i in count(0,2):
            i2 = 1<x:
                break
            for j in count(0):
                k = i2*3**j
                if k>x:
                    break
                c -= (x//k-1)//6+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

Formula

{a(n): n >= 1} = {m >= 1 : A007913(m) == 1 (mod 6)} U {m >= 1 : A007913(m) == 3 (mod 18)} = {A352272(m): m >= 1} U {3*A352272(m): m >= 1}.

{A225837(n): n >= 1} = {a(m): m >= 1} U {2*a(m): m >= 1}.

A225857 Numbers of the form 2^i3^j(12k+1) or 2^i3^j(12k+5), i, j, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 18, 20, 24, 25, 26, 27, 29, 30, 32, 34, 36, 37, 39, 40, 41, 45, 48, 49, 50, 51, 52, 53, 54, 58, 60, 61, 64, 65, 68, 72, 73, 74, 75, 77, 78, 80, 81, 82, 85, 87, 89, 90, 96, 97, 98, 100, 101, 102, 104, 106, 108, 109, 111

Offset: 1

Ralf Stephan, May 18 2013

From Peter Munn, Nov 11 2023: (Start)
Numbers k whose 5-rough part, A065330(k), is congruent to 1 modulo 4.
Contains all nonzero squares.
Positive integers in the multiplicative subgroup of rationals generated by 2, 3, 5 and integers congruent to 1 modulo 12. Thus, the sequence is closed under multiplication and, provided the result is an integer, under division.
This subgroup has index 2 and does not include -1, so is the complement of its negation. In respect of the sequence, the index 2 property implies we can take any absent positive integer m, and divide by m all terms that are multiples of m to get the complementary sequence, A225858.
Likewise, the sequence forms a subgroup of index 2 of the positive integers under the operation A059897(.,.).
(End)
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Nov 14 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A225858.

Cf. A059897, A065330, A225837.

[n: n in [1..200] | d mod 4 eq 1 where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // Bruno Berselli, May 16 2013

Select[Range[120], Mod[#/Times @@ ({2, 3}^IntegerExponent[#, {2, 3}]), 4] == 1 &] (* Amiram Eldar, Nov 14 2023 *)

for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%4==1),print1(n,",")))

from itertools import count
from sympy import integer_log
def A225857(n):
    def f(x):
        c = n
        for i in range(integer_log(x,3)[0]+1):
            i2 = 3**i
            for j in count(0):
                k = i2<x:
                    break
                m = x//k
                c += (m-7)//12+(m-11)//12+2
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 24 2025

Name clarified by Peter Munn, Nov 10 2023

cp's OEIS Frontend

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

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A225838 Numbers of form 2^i*3^j*(6k+5), i, j, k >= 0.

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A352272 Numbers whose squarefree part is congruent to 1 modulo 6.

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A352273 Numbers whose squarefree part is congruent to 5 modulo 6.

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Mathematica

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A352274 Numbers whose squarefree part is congruent to 1 modulo 6 or 3 modulo 18.

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Programs

PARI

Python

Formula

A225857 Numbers of the form 2^i*3^j*(12k+1) or 2^i*3^j*(12k+5), i, j, k >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Extensions

A036668 Hati numbers: of form 2^i3^jk, i+j even, (k,6)=1.

A225838 Numbers of form 2^i3^j(6k+5), i, j, k >= 0.

A225857 Numbers of the form 2^i3^j(12k+1) or 2^i3^j(12k+5), i, j, k >= 0.