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

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

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.

Programs

Mathematica

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 *)

PARI

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

Python

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

Formula

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)

Extensions

New name from Peter Munn, Mar 17 2022