login
a(n) = largest nondivisor k < n such that A007947(k) | n, or 0 if k does not exist.
3

%I #7 Jul 07 2024 20:55:23

%S 0,0,0,0,0,4,0,0,0,8,0,9,0,8,9,0,0,16,0,16,9,16,0,18,0,16,0,16,0,27,0,

%T 0,27,32,25,32,0,32,27,32,0,36,0,32,27,32,0,36,0,40,27,32,0,48,25,49,

%U 27,32,0,54,0,32,49,0,25,64,0,64,27,64,0,64,0,64,45

%N a(n) = largest nondivisor k < n such that A007947(k) | n, or 0 if k does not exist.

%C The number k does not exist for n in A000961, therefore we write a(n) = 0.

%C For n in A024619, a(n) is the largest term in row n of A162306 or A272618.

%C For n in A024619, a(n) is composite, since A007947(p) | n implies p | n for prime p.

%H Michael De Vlieger, <a href="/A373736/b373736.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A373736/a373736.png">Scalar scatterplot of a(n)</a> for n = 1..2^20.

%H Michael De Vlieger, <a href="/A373736/a373736_1.png">Plot a(n) at (x,y) = (n mod 210, -floor(n/210))</a> for n = 1..44100, showing 0 in light gray, perfect prime powers (a(n) in A246547) in gold, a(n) in A332785 in blue, and a(n) in A286708 in magenta.

%e Let rad = A007947 and let S(n) = {k <= n : rad(k) | n}, i.e., row n of A162306.

%e a(6) = 4 since 4 is the largest nondivisor k in S(6) = {1, 2, 3, 4, 6}.

%e a(10) = 8 since 8 is the largest nondivisor k in S(10) = {1, 2, 4, 5, 8, 10}.

%e a(15) = 9 since 9 is the largest nondivisor k in S(15) = {1, 3, 5, 9, 15}, etc.

%t rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];

%t Table[If[PrimePowerQ[n], 0, k = n - 1; Until[And[Divisible[n, rad[k]], ! Divisible[n, k]], k--]; k], {n, 2, 120}]

%o (PARI) rad(n) = factorback(factorint(n)[, 1]);

%o a(n) = forstep(k=n-1, 1, -1, if ((n % k) && !(n % rad(k)), return(k))); \\ _Michel Marcus_, Jun 18 2024

%Y Cf. A000961, A007947, A024619, A162306, A272618.

%K nonn

%O 1,6

%A _Michael De Vlieger_, Jun 18 2024