cp's OEIS Frontend

A361487 Odd numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).

%I A361487 #11 Apr 01 2023 13:28:59
%S A361487 75,135,147,189,225,245,363,375,405,441,507,525,567,605,675,735,825,
%T A361487 845,847,867,875,891,945,975,1029,1053,1083,1089,1125,1183,1215,1225,
%U A361487 1275,1323,1375,1377,1425,1445,1485,1521,1539,1575,1587,1617,1625,1701,1715,1725,1755,1805,1815,1859,1863,1875,1911
%N A361487 Odd numbers k that are neither prime powers nor squarefree, such that k/rad(k) >= q, where rad(k) = A007947(k) and prime q = A119288(k).
%C A361487 Odd terms in A360768, which itself is a proper subsequence of A126706.
%C A361487 Odd numbers k such that there exists j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.
%H A361487 Michael De Vlieger, <a href="/A361487/b361487.txt">Table of n, a(n) for n = 1..10000</a>
%H A361487 Michael De Vlieger, <a href="/A361487/a361487.png">1020 pixel square bitmap</a> of indices n = 1..1040400, read left to right, top to bottom, such that A360768(n) in this sequence appears in black, else white. There is a faint pattern apparently related to that mentioned in A360768.
%H A361487 Michael De Vlieger, <a href="/A361487/a361487_1.png">Chart showing k < a(n)</a>, n = 1..36, rows n contain k such that rad(k) = rad(n), yet k does not divide n. These k are in A360769, the number of k in row a(n) given by A355432(a(n)).
%F A361487 This sequence is { odd k in A126706 : k/A007947(k) >= A119288(k) }.
%e A361487 a(1) = 75, since 75/15 >= 5. We note that rad(45) = rad(75) = 15, yet 45 does not divide 75.
%e A361487 a(2) = 135, since 135/15 >= 5. Note: rad(75) = rad(135) = 15, yet 45 does not divide 135.
%e A361487 a(3) = 147, since 147/21 >= 7. Note: rad(63) = rad(147) = 21, yet 147 mod 63 = 21.
%e A361487 Chart below shows k < a(n) such that rad(k) = rad(n), yet k does not divide n:
%e A361487       75 | 45   .
%e A361487      135 |  .   .  75   .   .
%e A361487      147 |  .  63   .   .   .   .
%e A361487      189 |  .   .   .   .   .   . 147   .   .   .
%e A361487 a(n) 225 |  .   .   .   .   . 135   .   .   .   .   .   .
%e A361487      245 |  .   .   .   .   .   .   .   .   . 175   .   .   .
%e A361487      363 |  .   .   .  99   .   .   .   .   .   .   .   .   .   .   .   .   . 297
%e A361487      375 | 45   .   .   .   . 135   .   .   .   .   .   . 225   .   .   .   .   .
%e A361487      ----------------------------------------------------------------------------
%e A361487          | 45  63  75  99 117 135 147 153 171 175 189 207 225 245 261 275 279 297
%e A361487                                         k in A360769
%t A361487 Select[Select[Range[1, 2000, 2], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]
%o A361487 (PARI) is(k) = { if (k%2, my (f = factor(k)); #f~ > 1 && k/vecprod(f[,1]~) >= f[2, 1], 0); } \\ _Rémy Sigrist_, Mar 29 2023
%Y A361487 Cf. A005408, A007947, A013929, A024619, A119288, A126706, A355432, A360768, A360769.
%K A361487 nonn
%O A361487 1,1
%A A361487 _Michael De Vlieger_, Mar 29 2023