A373736 a(n) = largest nondivisor k < n such that A007947(k) | n, or 0 if k does not exist.

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

Offset: 1

Author

Michael De Vlieger, Jun 18 2024

Keywords

• nonn

Comments

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

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

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

Examples

			Let rad = A007947 and let S(n) = {k <= n : rad(k) | n}, i.e., row n of A162306.
a(6) = 4 since 4 is the largest nondivisor k in S(6) = {1, 2, 3, 4, 6}.
a(10) = 8 since 8 is the largest nondivisor k in S(10) = {1, 2, 4, 5, 8, 10}.
a(15) = 9 since 9 is the largest nondivisor k in S(15) = {1, 3, 5, 9, 15}, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Scalar scatterplot of a(n) for n = 1..2^20.
• Michael De Vlieger, Plot a(n) at (x,y) = (n mod 210, -floor(n/210)) 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.

Crossrefs

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

Programs

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[If[PrimePowerQ[n], 0, k = n - 1; Until[And[Divisible[n, rad[k]], ! Divisible[n, k]], k--]; k], {n, 2, 120}]

PARI

rad(n) = factorback(factorint(n)[, 1]);
a(n) = forstep(k=n-1, 1, -1, if ((n % k) && !(n % rad(k)), return(k))); \\ Michel Marcus, Jun 18 2024