cp's OEIS Frontend

A352272 Numbers whose squarefree part is congruent to 1 modulo 6.

%I A352272 #13 Feb 16 2025 08:34:03
%S A352272 1,4,7,9,13,16,19,25,28,31,36,37,43,49,52,55,61,63,64,67,73,76,79,81,
%T A352272 85,91,97,100,103,109,112,115,117,121,124,127,133,139,144,145,148,151,
%U A352272 157,163,169,171,172,175,181,187,193,196,199,205,208,211,217,220,223,225,229
%N A352272 Numbers whose squarefree part is congruent to 1 modulo 6.
%C A352272 Numbers of the form 4^i * 9^j * (6k+1), i, j, k >= 0.
%C A352272 Closed under multiplication.
%C A352272 The sequence forms a subgroup of the positive integers under the commutative operation A059897(.,.), one of 8 subgroups of the form {k : A007913(k) == 1 (mod m)} - in each case m is a divisor of 24. A059897 has a relevance to squarefree parts that arises from the identity A007913(k*n) = A059897(A007913(k), A007913(n)), where A007913(n) is the squarefree part of n.
%C A352272 The subgroup has 8 cosets, which partition the positive integers as follows. For each i in {1, 5}, j in {1, 2, 3, 6} there is a coset {m^2 * (6k+i) * j : m >= 1, k >= 0}. See the table in the examples.
%C A352272 None of the 8 cosets have been entered into the database previously, but many subgroups of the quotient group (which are formed of certain combinations of cosets) are represented among earlier OEIS sequences, including 6 of the 7 subgroups of index 2 (which combine 4 cosets). This sequence can therefore be defined as the intersection of pairs or triples of these sequences in many combinations (see the cross-references). See also the table in the example section of A352273 (the coset that includes 5).
%H A352272 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Group.html">Group</a>, <a href="https://mathworld.wolfram.com/QuotientGroup.html">Quotient Group</a>, <a href="https://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.
%F A352272 {a(n) : n >= 1} = {m >= 1 : A007913(m) == 1 (mod 6)}.
%F A352272 {a(n) : n >= 1} = A334832 U A334832/7 U A334832/13 U A334832/19 where A334832/k denotes {A334832(m)/k : m >= 1, k divides A334832(m)}.
%F A352272 Using the same denotation, {a(n) : n >= 1} = A352273/5 = {A307151(A352273(m)) : m >= 1}.
%e A352272 The squarefree part of 9 is 1, which is congruent to 1 (mod 6), so 9 is in the sequence.
%e A352272 The squarefree part of 14 is 14, which is congruent to 2 (mod 6), so 14 is not in the sequence.
%e A352272 The squarefree part of 52 = 2^2 * 13 is 13, which is congruent to 1 (mod 6), so 52 is in the sequence.
%e A352272 The 8 cosets described in the initial comments (forming a partition of the positive integers) are shown as rows of the following table. The first half of the table corresponds to (6k+i) with i=1; the second half to i=5, with row 5 being A352273.
%e A352272    1,  4,   7,   9,  13,  16,  19,  25,  28,  31,  36, ...
%e A352272    2,  8,  14,  18,  26,  32,  38,  50,  56,  62,  72, ...
%e A352272    3, 12,  21,  27,  39,  48,  57,  75,  84,  93, 108, ...
%e A352272    6, 24,  42,  54,  78,  96, 114, 150, 168, 186, 216, ...
%e A352272    5, 11,  17,  20,  23,  29,  35,  41,  44,  45,  47, ...
%e A352272   10, 22,  34,  40,  46,  58,  70,  82,  88,  90,  94, ...
%e A352272   15, 33,  51,  60,  69,  87, 105, 123, 132, 135, 141, ...
%e A352272   30, 66, 102, 120, 138, 174, 210, 246, 264, 270, 282, ...
%e A352272 The product of two positive integers is in this sequence if and only if they are in the same coset. The asymptotic density of cosets (containing) 1 and 5 is 1/4; of cosets 2 and 10 is 1/8; of cosets 3 and 15 is 1/12; of cosets 6 and 30 is 1/24.
%o A352272 (PARI) isok(m) = core(m) % 6 == 1;
%o A352272 (Python)
%o A352272 from itertools import count
%o A352272 def A352272(n):
%o A352272     def bisection(f,kmin=0,kmax=1):
%o A352272         while f(kmax) > kmax: kmax <<= 1
%o A352272         kmin = kmax >> 1
%o A352272         while kmax-kmin > 1:
%o A352272             kmid = kmax+kmin>>1
%o A352272             if f(kmid) <= kmid:
%o A352272                 kmax = kmid
%o A352272             else:
%o A352272                 kmin = kmid
%o A352272         return kmax
%o A352272     def f(x):
%o A352272         c = n+x
%o A352272         for i in count(0):
%o A352272             i2 = 9**i
%o A352272             if i2>x:
%o A352272                 break
%o A352272             for j in count(0,2):
%o A352272                 k = i2<<j
%o A352272                 if k>x:
%o A352272                     break
%o A352272                 c -= (x//k-1)//6+1
%o A352272         return c
%o A352272     return bisection(f,n,n) # _Chai Wah Wu_, Feb 14 2025
%Y A352272 Intersection of any 2 of A055047, A339690 and A352274.
%Y A352272 Intersection of any 4 sets chosen from A003159, A007417, A026225, A036668, A189715 and A225837 (in most cases, only 3 sets are needed - specifically if the pairwise intersections of the 3 sets differ from each other).
%Y A352272 Closure of A084089 under multiplication by 9.
%Y A352272 Other subsequences: A000290\{0}, A016921, A229848 (apparently, with 55 the first difference).
%Y A352272 A334832 lists equivalent sequences modulo other divisors of 24.
%Y A352272 Cf. A007913, A059897, A307151, A352273.
%K A352272 nonn,easy
%O A352272 1,2
%A A352272 _Peter Munn_, Mar 10 2022