A304572 - OEIS

%I #13 Mar 18 2026 03:48:56

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

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

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

%N Triangle read by rows: T(n,k) = 1 if k does not divide n^e, positive nonzero integers, and gcd(n,k) > 1.

%C T(n,k) = 1 iff both A304569(n,k) = 0 and A054521(n,k) = 0; T(n,k) = 0 otherwise.

%C This sequence contains 1 where 1 appears in row n of A304571 but not A304569.

%C Row n of A272619 contains indices of 1 in this sequence.

%C A243823(n) = total of row n in this sequence.

%C Rows n for n prime and n <= 6 contain only zeros; all other rows have at least one 1.

%C T(n,k) = 0 for k prime.

%H Michael De Vlieger, <a href="/A304572/b304572.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150)

%H Michael De Vlieger, <a href="/A304572/a304572.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, 0, 0, 0;

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

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

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

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

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

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

%e ...

%t Table[Array[Boole[And[PowerMod[n, Floor@ Log2@ n, #] != 0, GCD[n, #] > 1]] &, n], {n, 13}] // Flatten

%t (* Alternative: extended data in rows from PNG image above: first, download the PNG and save it as "a304572.png", provides 2669205 terms: *)

%t MapIndexed[Take[#1, First@ #2] &, ImageData@ ColorNegate@ Import["a304572.png", "PNG"]] (* _Michael De Vlieger_, Jul 02 2018 *)

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

%Y Cf. A054521, A243823, A272619, A304569, A304571.

%K nonn,easy,tabl

%O 1

%A _Michael De Vlieger_, May 23 2018