A352274 -id:A352274 - cp's OEIS frontend

A026225 Numbers of the form 3^i * (3k+1).

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 30, 31, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 55, 57, 58, 61, 63, 64, 66, 67, 70, 73, 75, 76, 79, 81, 82, 84, 85, 88, 90, 91, 93, 94, 97, 100, 102, 103, 106, 108, 109, 111, 112, 115

Offset: 1

Clark Kimberling

Old name: a(n) = (1/3)*(s(n+1) - 1), where s = A026224.
Conjectures based on old name: these are numbers of the form (3*i+1)*3^j; see A182828, and they comprise the complement of A026179, except for the initial 1 in A026179.
From Peter Munn, Mar 17 2022: (Start)
Numbers with an even number of prime factors of the form 3k-1 counting repetitions.
Numbers whose squarefree part is congruent to 1 modulo 3 or 3 modulo 9.
The integers in an index 2 subgroup of the positive rationals under multiplication. As such the sequence is closed under multiplication and - where the result is an integer - under division; also for any positive integer k not in the sequence, the sequence's complement is generated by dividing by k the terms that are multiples of k.
Alternatively, the sequence can be viewed as an index 2 subgroup of the positive integers under the commutative binary operation A059897(.,.).
Viewed either way, the sequence corresponds to a subgroup of the quotient group derived in the corresponding way from A055047. (End)
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022
Is this A026140 shifted right? - R. J. Mathar, Jun 24 2025

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

Elements of array A182828 in ascending order.
Union of A055041 and A055047.
Other subsequences: A007645 (primes), A352274.
Symmetric difference of A003159 and A225838; of A007417 and A189716.
Cf. A001222, A059897, A343430.

a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, 160}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[3, 1]  (* A026225 *)
p[3, 2] (* A026179 without initial 1 *)
(* Clark Kimberling, Oct 19 2016 *)

isok(m) = core(m) % 3 == 1 || core(m) % 9 == 3; \\ Peter Munn, Mar 17 2022

from sympy import integer_log
def A026225(n):
    def f(x): return n+x-sum(((x//3**i)-1)//3+1 for i in range(integer_log(x,3)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 15 2025

From Peter Munn, Mar 17 2022: (Start)
{a(n) : n >= 1} = {m : A001222(A343430(m)) == 0 (mod 2)}.
{a(n) : n >= 1} = {A055047(m) : m >= 1} U {3*A055047(m) : m >= 1}.
{a(n) : n >= 1} = {A352274(m) : m >= 1} U {A352274(m)/10 : m >= 1, 10 divides A352274(m)}. (End)

New name from Peter Munn, Mar 17 2022

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}.

A084089 Numbers k such that k == 1 (mod 3) and the exponent of the highest power of 2 dividing k is even.

Original entry on oeis.org

1, 4, 7, 13, 16, 19, 25, 28, 31, 37, 43, 49, 52, 55, 61, 64, 67, 73, 76, 79, 85, 91, 97, 100, 103, 109, 112, 115, 121, 124, 127, 133, 139, 145, 148, 151, 157, 163, 169, 172, 175, 181, 187, 193, 196, 199, 205, 208, 211, 217, 220, 223, 229, 235

Offset: 1

Ralf Stephan, May 11 2003

Numbers that are both in A016777 and A003159.
It seems that lim_{n->oo} a(n)/n = 9/2. [This is true. The asymptotic density of this sequence is 2/9. - Amiram Eldar, Jan 16 2022]
Positions of +1 in the expansion of Sum_{k>=0} x^2^k/(1+x^2^k+x^2^(k+1)) (A084091).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for 2-automatic sequences.

Intersection of A003159 and A016777.
Cf. A084091.
A352274 without the multiples of 3.

Select[3 * Range[0, 81] + 1, EvenQ[IntegerExponent[#, 2]] &] (* Amiram Eldar, Jan 16 2022 *)

for(n=0,300,if(valuation(n,2)%2==0&&n%3==1,print1(n",")))

from itertools import count, islice
def A084089_gen(): # generator of terms
    return filter(lambda n:(n&-n).bit_length()&1,count(1,3))
A084089_list = list(islice(A084089_gen(),30)) # Chai Wah Wu, Jul 11 2022

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}.

cp's OEIS Frontend

A026225 Numbers of the form 3^i * (3k+1).

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

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A084089 Numbers k such that k == 1 (mod 3) and the exponent of the highest power of 2 dividing k is even.

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula