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).

75, 135, 147, 189, 225, 245, 363, 375, 405, 441, 507, 525, 567, 605, 675, 735, 825, 845, 847, 867, 875, 891, 945, 975, 1029, 1053, 1083, 1089, 1125, 1183, 1215, 1225, 1275, 1323, 1375, 1377, 1425, 1445, 1485, 1521, 1539, 1575, 1587, 1617, 1625, 1701, 1715, 1725, 1755, 1805, 1815, 1859, 1863, 1875, 1911

Offset: 1

Author

Michael De Vlieger, Mar 29 2023

Keywords

• nonn

Comments

Odd terms in A360768, which itself is a proper subsequence of A126706.

Odd numbers k such that there exists j such that 1 < j < k and rad(j) = rad(k), but j does not divide k.

Examples

			a(1) = 75, since 75/15 >= 5. We note that rad(45) = rad(75) = 15, yet 45 does not divide 75.
a(2) = 135, since 135/15 >= 5. Note: rad(75) = rad(135) = 15, yet 45 does not divide 135.
a(3) = 147, since 147/21 >= 7. Note: rad(63) = rad(147) = 21, yet 147 mod 63 = 21.
Chart below shows k < a(n) such that rad(k) = rad(n), yet k does not divide n:
      75 | 45   .
     135 |  .   .  75   .   .
     147 |  .  63   .   .   .   .
     189 |  .   .   .   .   .   . 147   .   .   .
a(n) 225 |  .   .   .   .   . 135   .   .   .   .   .   .
     245 |  .   .   .   .   .   .   .   .   . 175   .   .   .
     363 |  .   .   .  99   .   .   .   .   .   .   .   .   .   .   .   .   . 297
     375 | 45   .   .   .   . 135   .   .   .   .   .   . 225   .   .   .   .   .
     ----------------------------------------------------------------------------
         | 45  63  75  99 117 135 147 153 171 175 189 207 225 245 261 275 279 297
                                        k in A360769
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, 1020 pixel square bitmap 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.
• Michael De Vlieger, Chart showing k < a(n), 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)).

Crossrefs

Cf. A005408, A007947, A013929, A024619, A119288, A126706, A355432, A360768, A360769.

Programs

Mathematica

Select[Select[Range[1, 2000, 2], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]

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

Formula

This sequence is { odd k in A126706 : k/A007947(k) >= A119288(k) }.