cp's OEIS Frontend

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

%I A026225 #31 Jun 24 2025 13:16:10
%S A026225 1,3,4,7,9,10,12,13,16,19,21,22,25,27,28,30,31,34,36,37,39,40,43,46,
%T A026225 48,49,52,55,57,58,61,63,64,66,67,70,73,75,76,79,81,82,84,85,88,90,91,
%U A026225 93,94,97,100,102,103,106,108,109,111,112,115
%N A026225 Numbers of the form 3^i * (3k+1).
%C A026225 Old name: a(n) = (1/3)*(s(n+1) - 1), where s = A026224.
%C A026225 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.
%C A026225 From _Peter Munn_, Mar 17 2022: (Start)
%C A026225 Numbers with an even number of prime factors of the form 3k-1 counting repetitions.
%C A026225 Numbers whose squarefree part is congruent to 1 modulo 3 or 3 modulo 9.
%C A026225 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.
%C A026225 Alternatively, the sequence can be viewed as an index 2 subgroup of the positive integers under the commutative binary operation A059897(.,.).
%C A026225 Viewed either way, the sequence corresponds to a subgroup of the quotient group derived in the corresponding way from A055047. (End)
%C A026225 The asymptotic density of this sequence is 1/2. - _Amiram Eldar_, Apr 03 2022
%C A026225 Is this A026140 shifted right? - _R. J. Mathar_, Jun 24 2025
%H A026225 Amiram Eldar, <a href="/A026225/b026225.txt">Table of n, a(n) for n = 1..10000</a>
%H A026225 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.
%F A026225 From _Peter Munn_, Mar 17 2022: (Start)
%F A026225 {a(n) : n >= 1} = {m : A001222(A343430(m)) == 0 (mod 2)}.
%F A026225 {a(n) : n >= 1} = {A055047(m) : m >= 1} U {3*A055047(m) : m >= 1}.
%F A026225 {a(n) : n >= 1} = {A352274(m) : m >= 1} U {A352274(m)/10 : m >= 1, 10 divides A352274(m)}. (End)
%t A026225 a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, 160}]
%t A026225 p[b_, d_] := Flatten[Position[a[b], d]]
%t A026225 p[3, 1]  (* A026225 *)
%t A026225 p[3, 2] (* A026179 without initial 1 *)
%t A026225 (* _Clark Kimberling_, Oct 19 2016 *)
%o A026225 (PARI) isok(m) = core(m) % 3 == 1 || core(m) % 9 == 3; \\ _Peter Munn_, Mar 17 2022
%o A026225 (Python)
%o A026225 from sympy import integer_log
%o A026225 def A026225(n):
%o A026225     def f(x): return n+x-sum(((x//3**i)-1)//3+1 for i in range(integer_log(x,3)[0]+1))
%o A026225     m, k = n, f(n)
%o A026225     while m != k: m, k = k, f(k)
%o A026225     return m # _Chai Wah Wu_, Feb 15 2025
%Y A026225 Elements of array A182828 in ascending order.
%Y A026225 Union of A055041 and A055047.
%Y A026225 Other subsequences: A007645 (primes), A352274.
%Y A026225 Symmetric difference of A003159 and A225838; of A007417 and A189716.
%Y A026225 Cf. A001222, A059897, A343430.
%K A026225 nonn,easy
%O A026225 1,2
%A A026225 _Clark Kimberling_
%E A026225 New name from _Peter Munn_, Mar 17 2022