A304570 - OEIS

%I #13 Nov 09 2018 21:33:17

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0

%N Triangle read by rows: T(n,k) = 1 if k | n^e with e > 1 and n mod k != 0, T(n,k) = 0 otherwise (1 <= k <= n).

%C T(n,k) = 1 iff A051731(n,k) = 0 but A304569(n,k) = 1; T(n,k) = 0 otherwise. This sequence contains 1 where 1 appears in row n of A304569 but not in same row of A051731.

%C Row n of A272618 contains indices of 1 in this sequence.

%C A243822(n) = total of row n in this sequence.

%C Row n such that omega(n) = 1 contains zeros.

%H Michael De Vlieger, <a href="/A304570/b304570.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150)

%H Michael De Vlieger, <a href="/A304570/a304570.png">Image of rows 1 <= n <= 2310</a>

%e Table begins:

%e 0;

%e 0, 0;

%e 0, 0, 0;

%e 0, 0, 0, 0;

%e 0, 0, 0, 0, 0;

%e 0, 0, 0, 1, 0, 0;

%e 0, 0, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0;

%e 0, 0, 0, 1, 0, 0, 0, 1, 0, 0;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0,;

%e ...

%t Table[Boole[And[PowerMod[n, Floor@ Log2@ n, #] == 0, Mod[n, #] > 0]], {n, 13}] // Flatten

%t (* Second program: extended data in rows from PNG image above: first, download the PNG and save it as "a304570.png", provides 2669205 terms: *)

%t MapIndexed[Take[#1, First@ #2] &, ImageData@ ColorNegate@ Import["a304570.png", "PNG"]] (* _Michael De Vlieger_, Jul 02 2018 *)

%o (PARI) T(n,k) = {my(r=vecprod(factor(k)[,1])); n%k<>0 && n%r==0} \\ _Andrew Howroyd_, Nov 08 2018

%Y Cf. A051731, A162306, A243822, A272618, A304569.

%K nonn,easy,tabl

%O 1

%A _Michael De Vlieger_, May 23 2018

%E Name clarified by _Andrew Howroyd_, Nov 08 2018