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.
%I A352273 #17 Feb 16 2025 08:34:03
%S A352273 5,11,17,20,23,29,35,41,44,45,47,53,59,65,68,71,77,80,83,89,92,95,99,
%T A352273 101,107,113,116,119,125,131,137,140,143,149,153,155,161,164,167,173,
%U A352273 176,179,180,185,188,191,197,203,207,209,212,215,221,227,233,236,239,245,251
%N A352273 Numbers whose squarefree part is congruent to 5 modulo 6.
%C A352273 Numbers of the form 4^i * 9^j * (6k+5), i, j, k >= 0.
%C A352273 1/5 of each multiple of 5 in A352272.
%C A352273 The product of any two terms is in A352272.
%C A352273 The product of a term of this sequence and a term of A352272 is a term of this sequence.
%C A352273 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.
%C A352273 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.
%C A352273 The asymptotic density of this sequence is 1/4. - _Amiram Eldar_, Apr 03 2022
%H A352273 Amiram Eldar, <a href="/A352273/b352273.txt">Table of n, a(n) for n = 1..10000</a>
%H A352273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.
%F A352273 {a(n) : n >= 1} = {m >= 1 : A007913(m) == 5 (mod 6)}.
%F A352273 {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)}.
%F A352273 Using the same notation, {a(n) : n >= 1} = A352272/5 = {A307151(A352272(m)) : m >= 1}.
%F A352273 {A225838(n) : n >= 1} = {m : m = a(j)*k, j >= 1, k divides 6}.
%e A352273 The squarefree part of 11 is 11, which is congruent to 5 (mod 6), so 11 is in the sequence.
%e A352273 The squarefree part of 15 is 15, which is congruent to 3 (mod 6), so 15 is not in the sequence.
%e A352273 The squarefree part of 20 = 2^2 * 5 is 5, which is congruent to 5 (mod 6), so 20 is in the sequence.
%e A352273 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.
%e A352273 H 2H 3H 6H H/5 2H/5 3H/5 6H/5
%e A352273 A003159 X X X X
%e A352273 A036554 X X X X
%e A352273 .
%e A352273 A007417 X X X X
%e A352273 A145204\{0} X X X X
%e A352273 .
%e A352273 A026225 X X X X
%e A352273 A026179\{1} X X X X
%e A352273 .
%e A352273 A036668 X X X X
%e A352273 A325424 X X X X
%e A352273 .
%e A352273 A055047 X X
%e A352273 A055048 X X
%e A352273 A055041 X X
%e A352273 A055040 X X
%e A352273 .
%e A352273 A189715 X X X X
%e A352273 A189716 X X X X
%e A352273 .
%e A352273 A225837 X X X X
%e A352273 A225838 X X X X
%e A352273 .
%e A352273 A339690 X X
%e A352273 A329575 X X
%e A352273 .
%e A352273 A352274 X X
%e A352273 (The sequence groupings in the table start with the subgroup of the quotient group of H, followed by its cosets.)
%t A352273 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 *)
%o A352273 (PARI) isok(m) = core(m) % 6 == 5;
%o A352273 (Python)
%o A352273 from itertools import count
%o A352273 def A352273(n):
%o A352273 def bisection(f,kmin=0,kmax=1):
%o A352273 while f(kmax) > kmax: kmax <<= 1
%o A352273 kmin = kmax >> 1
%o A352273 while kmax-kmin > 1:
%o A352273 kmid = kmax+kmin>>1
%o A352273 if f(kmid) <= kmid:
%o A352273 kmax = kmid
%o A352273 else:
%o A352273 kmin = kmid
%o A352273 return kmax
%o A352273 def f(x):
%o A352273 c = n+x
%o A352273 for i in count(0):
%o A352273 i2 = 9**i
%o A352273 if i2>x: break
%o A352273 for j in count(0,2):
%o A352273 k = i2<<j
%o A352273 if k>x: break
%o A352273 c -= (x//k-5)//6+1
%o A352273 return c
%o A352273 return bisection(f,n,n) # _Chai Wah Wu_, Feb 14 2025
%Y A352273 Intersection of any three of A003159, A007417, A189716 and A225838.
%Y A352273 Intersection of A036668 and A055048.
%Y A352273 Complement within A339690 of A352272.
%Y A352273 Cf. A007913, A059897, A307151, A334832.
%Y A352273 Closure of A084088 under multiplication by 9.
%Y A352273 Other subsequences: A033429\{0}, A016969.
%Y A352273 Other sequences in the example table: A036554, A145204, A026179, A026225, A325424, A055040, A055041, A055047, A189715, A225837, A329575, A352274.
%K A352273 nonn,easy
%O A352273 1,1
%A A352273 _Peter Munn_, Mar 10 2022