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

%I A352274 #13 Feb 16 2025 08:34:03
%S A352274 1,3,4,7,9,12,13,16,19,21,25,27,28,31,36,37,39,43,48,49,52,55,57,61,
%T A352274 63,64,67,73,75,76,79,81,84,85,91,93,97,100,103,108,109,111,112,115,
%U A352274 117,121,124,127,129,133,139,144,145,147,148,151,156,157,163,165,169,171,172
%N A352274 Numbers whose squarefree part is congruent to 1 modulo 6 or 3 modulo 18.
%C A352274 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).
%C A352274 Closed under multiplication.
%C A352274 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.
%C A352274 Term by term, the sequence is one half of its complement within A225837.
%H A352274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.
%F A352274 {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}.
%F A352274 {A225837(n): n >= 1} = {a(m): m >= 1} U {2*a(m): m >= 1}.
%e A352274 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.
%e A352274 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.
%e A352274 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.
%e A352274 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.
%o A352274 (PARI) isok(m) = core(m) % 6 == 1 || core(m) % 18 == 3;
%o A352274 (Python)
%o A352274 from itertools import count
%o A352274 def A352274(n):
%o A352274     def bisection(f,kmin=0,kmax=1):
%o A352274         while f(kmax) > kmax: kmax <<= 1
%o A352274         kmin = kmax >> 1
%o A352274         while kmax-kmin > 1:
%o A352274             kmid = kmax+kmin>>1
%o A352274             if f(kmid) <= kmid:
%o A352274                 kmax = kmid
%o A352274             else:
%o A352274                 kmin = kmid
%o A352274         return kmax
%o A352274     def f(x):
%o A352274         c = n+x
%o A352274         for i in count(0,2):
%o A352274             i2 = 1<<i
%o A352274             if i2>x:
%o A352274                 break
%o A352274             for j in count(0):
%o A352274                 k = i2*3**j
%o A352274                 if k>x:
%o A352274                     break
%o A352274                 c -= (x//k-1)//6+1
%o A352274         return c
%o A352274     return bisection(f,n,n) # _Chai Wah Wu_, Feb 14 2025
%Y A352274 Intersection of any two of A003159, A026225 and A225837.
%Y A352274 Closure of A084089 under multiplication by 3.
%Y A352274 Cf. A007913.
%Y A352274 Subsequences: A003136\{0}, A108166, A352272.
%K A352274 nonn,easy
%O A352274 1,2
%A A352274 _Peter Munn_, Mar 10 2022