A334832 - OEIS

%I #36 Mar 31 2022 12:58:29

%S 1,4,9,16,25,36,49,64,73,81,97,100,121,144,145,169,193,196,217,225,

%T 241,256,265,289,292,313,324,337,361,385,388,400,409,433,441,457,481,

%U 484,505,529,553,576,577,580,601,625,649,657,673,676,697,721,729,745,769,772,784,793,817,841

%N Numbers whose squarefree part is congruent to 1 (mod 24).

%C Closed under multiplication.

%C The sequence forms a subgroup of the positive integers under the commutative operation A059897(.,.). A059897 has a relevance to squarefree parts that arises from the identity A007913(k*m) = A059897(A007913(k), A007913(m)), where A007913(n) is the squarefree part of n.

%C The subgroup is one of 8 A059897(.,.) subgroups of the form {k : A007913(k) == 1 (mod m)}. The list seems complete, in anticipation of proof that such sets form subgroups only when m is a divisor of 24 (based on the property described by A. G. Astudillo in A018253). This sequence might be viewed as primitive with respect to the other 7, as the latter correspond to subgroups of its quotient group, in the sense that each one (as a set) is also a union of cosets described below. The 7 include A003159 (m=2), A055047 (m=3), A277549 (m=4), A234000 (m=8) and the trivial A000027 (m=1).

%C The subgroup has 32 cosets. For each i in {1, 5, 7, 11, 13, 17, 19, 23}, j in {1, 2, 3, 6} there is a coset {n : n = k^2 * (24m + i) * j, k >= 1, m >= 0}. The divisors of 2730 = 2*3*5*7*13 form a transversal. (11, clearly not such a divisor, is in the same coset as 35 = 11 + 24; 17, 19, 23 are in the same cosets as 65, 91, 455 respectively.)

%C The asymptotic density of this sequence is 1/16. - _Amiram Eldar_, Mar 08 2021

%H Amiram Eldar, <a href="/A334832/b334832.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F {a(n)} = {n : n = k^2 * (24m + 1), k >= 1, m >= 0}.

%e The squarefree part of 26 is 26, which is congruent to 2 (mod 24), so 26 is not in the sequence.

%e The squarefree part of 292 = 2^2 * 73 is 73, which is congruent to 1 (mod 24), so 292 is in the sequence.

%t Select[Range[850], Mod[Sqrt[#] /. (c_ : 1)*a_^(b_ : 0) :> (c*a^b)^2, 24] == 1 &] (* _Michael De Vlieger_, Jun 24 2020 *)

%o (PARI) isok(m) = core(m) % 24 == 1; \\  _Peter Munn_, Jun 21 2020

%Y A subgroup under A059897, defined using A007913.

%Y Subsequences: A000290\{0}, A103214, A107008.

%Y Equivalent sequences modulo other members of A018253: A000027 (1), A003159 (2), A055047 (3), A277549 (4), A352272(6), A234000 (8).

%K nonn,easy

%O 1,2

%A _Peter Munn_, Jun 15 2020