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A334832 Numbers whose squarefree part is congruent to 1 (mod 24).

%I A334832 #44 Apr 22 2025 05:08:51
%S A334832 1,4,9,16,25,36,49,64,73,81,97,100,121,144,145,169,193,196,217,225,
%T A334832 241,256,265,289,292,313,324,337,361,385,388,400,409,433,441,457,481,
%U A334832 484,505,529,553,576,577,580,601,625,649,657,673,676,697,721,729,745,769,772,784,793,817,841
%N A334832 Numbers whose squarefree part is congruent to 1 (mod 24).
%C A334832 Closed under multiplication.
%C A334832 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 A334832 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 A334832 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 A334832 The asymptotic density of this sequence is 1/16. - _Amiram Eldar_, Mar 08 2021
%H A334832 Amiram Eldar, <a href="/A334832/b334832.txt">Table of n, a(n) for n = 1..10000</a>
%H A334832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuotientGroup.html">Quotient Group</a>.
%H A334832 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RightTransversal.html">Right Transversal</a>.
%F A334832 {a(n)} = {n : n = k^2 * (24m + 1), k >= 1, m >= 0}.
%e A334832 The squarefree part of 26 is 26, which is congruent to 2 (mod 24), so 26 is not in the sequence.
%e A334832 The squarefree part of 292 = 2^2 * 73 is 73, which is congruent to 1 (mod 24), so 292 is in the sequence.
%t A334832 Select[Range[850], Mod[Sqrt[#] /. (c_ : 1)*a_^(b_ : 0) :> (c*a^b)^2, 24] == 1 &] (* _Michael De Vlieger_, Jun 24 2020 *)
%o A334832 (PARI) isok(m) = core(m) % 24 == 1; \\ _Peter Munn_, Jun 21 2020
%o A334832 (Python)
%o A334832 from sympy import integer_log
%o A334832 def A334832(n):
%o A334832     def bisection(f,kmin=0,kmax=1):
%o A334832         while f(kmax) > kmax: kmax <<= 1
%o A334832         kmin = kmax >> 1
%o A334832         while kmax-kmin > 1:
%o A334832             kmid = kmax+kmin>>1
%o A334832             if f(kmid) <= kmid:
%o A334832                 kmax = kmid
%o A334832             else:
%o A334832                 kmin = kmid
%o A334832         return kmax
%o A334832     def f(x): return n+x-sum((x//(i<<(j<<1))-1)//24+1 for i in (9**k for k in range(integer_log(x,9)[0]+1)) for j in range((x//i>>1).bit_length()+1))
%o A334832     return bisection(f,n,n) # _Chai Wah Wu_, Mar 21 2025
%Y A334832 A subgroup under A059897, defined using A007913.
%Y A334832 Subsequences: A000290\{0}, A103214, A107008.
%Y A334832 Equivalent sequences modulo other members of A018253: A000027 (1), A003159 (2), A055047 (3), A277549 (4), A352272(6), A234000 (8).
%K A334832 nonn,easy
%O A334832 1,2
%A A334832 _Peter Munn_, Jun 15 2020