A108861 Numbers k that divide the sum of the digits of 2^k * k!.

1, 2, 3, 5, 6, 9, 27, 81, 126, 159, 205, 252, 254, 267, 285, 675, 1053, 1086, 1125, 1146, 2007, 5088, 5382, 5448, 14652, 23401, 23574, 24009, 41004, 66789, 67482, 111480, 866538, 1447875, 2413152, 2414019, 2417828, 2421360, 4045482, 6713982

Offset: 1

Author

Ryan Propper, Jul 11 2005

Keywords

• nonn
• base
• hard
• more

Comments

Next term after 5448 is greater than 10000.

a(34) > 10^6. - D. S. McNeil, Mar 03 2009

a(39) > 2.54 * 10^6, if it exists. - Kevin P. Thompson, Oct 20 2021

a(41) > 7*10^6, if it exists. - Kevin P. Thompson, Dec 08 2021

Examples

			9 is a term because the sum of the digits of 2^9 * 9! = 185794560 is 45 which is divisible by 9.
		

Crossrefs

Cf. A000165, A007953, A052582.

Programs

Mathematica

Do[If[Mod[Plus @@ IntegerDigits[2^n * n! ], n] == 0, Print[n]], {n, 1, 10000}]
Select[Range[6714000],Mod[Total[IntegerDigits[2^# #!]],#]==0&] (* Harvey P. Dale, Jul 11 2023 *)

PARI

isok(k) = !(sumdigits(2^k * k!) % k); \\ Michel Marcus, Oct 20 2021

Python

from itertools import islice
def A108861(): # generator of terms
    k, k2, kf = 1, 2, 1
    while True:
        c = sum(int(d) for d in str(k2*kf))
        if not c % k: yield k
        k += 1
        k2 *= 2
        kf *= k
A108861_list = list(islice(A108861(),10)) # Chai Wah Wu, Oct 26 2021

Extensions

a(25)-a(33) from D. S. McNeil, Mar 03 2009

a(34)-a(38) from Kevin P. Thompson, Oct 20 2021

a(39)-a(40) from Kevin P. Thompson, Dec 08 2021