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a(n) is the least number k such that k! is divisible by (k+1)^n but not by (k+1)^(n+1).
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%I #39 Jun 10 2019 06:07:38

%S 1,5,14,17,11,31,23,35,39,44,47,99,83,59,153,164,71,95,79,125,89,134,

%T 285,199,311,263,167,119,296,188,159,329,543,209,143,223,299,384,395,

%U 323,251,679,349,179,279,747,571,485,399,404,314,527,319,335,449,511,287,239,714

%N a(n) is the least number k such that k! is divisible by (k+1)^n but not by (k+1)^(n+1).

%C k+1 is not a prime.

%C a(n) + 1 is 17-smooth in DATA. - _David A. Corneth_, Mar 15 2019

%C But fails at n 99, 114, 125, 127, 130, 135, 143, 146, ... - _Michel Marcus_, Apr 30 2019

%F a(n) = A133481(n+1) - 1.

%F a(n) >= A061768(n).

%F If n = floor((p^j-1)/(j*(p-1)))-1, a(n) <= p^j-1 for prime p. For example, (p = 2), a(n) <= 2^j-1 for n = floor((2^j-1)/j)-1 (A082482(j)-1).

%e For n = 1, 1! = 1 is not divisible by 2, 2! = 2 is not divisible by 3, 3! = 6 is not divisible by 4, 4! = 24 is not divisible by 5, and 5! = 120 is divisible by 6 but not 36. Therefore a(1) = 5. - _Michael B. Porter_, Apr 21 2019

%t Array[Block[{k = 1}, While[Nand[Mod[k!, (k + 1)^#] == 0, Mod[k!, (k + 1)^(# + 1)] != 0], k++]; k] &, 58] (* _Michael De Vlieger_, Mar 11 2019 *)

%o (PARI) a(n) = {my(k=1); while((k! % (k+1)^n) || !(k! % (k+1)^(n+1)), k++); k; }

%Y Cf. A061768, A082482, A133481, A240751.

%K nonn

%O 0,2

%A _Jinyuan Wang_, Mar 09 2019