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Numbers whose squarefree part is congruent to 1 modulo 6 or 3 modulo 18.
4

%I #7 Mar 18 2022 12:28:06

%S 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 63,64,67,73,75,76,79,81,84,85,91,93,97,100,103,108,109,111,112,115,

%U 117,121,124,127,129,133,139,144,145,147,148,151,156,157,163,165,169,171,172

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

%C 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 Closed under multiplication.

%C 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 Term by term, the sequence is one half of its complement within A225837.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.

%F {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 {A225837(n): n >= 1} = {a(m): m >= 1} U {2*a(m): m >= 1}.

%e 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 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 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 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 (PARI) isok(m) = core(m) % 6 == 1 || core(m) % 18 == 3;

%Y Intersection of any two of A003159, A026225 and A225837.

%Y Closure of A084089 under multiplication by 3.

%Y Cf. A007913.

%Y Subsequences: A003136\{0}, A108166, A352272.

%K nonn,easy

%O 1,2

%A _Peter Munn_, Mar 10 2022