A138572 Numbers k that divide the sum of the digits of k^k in base 2.

1, 6, 122, 2126, 7910, 8254, 16201, 32312, 32426, 32998, 65436, 261649, 261803, 1044017, 1050183, 4194999

Offset: 1

Author

Robert Gerbicz, May 12 2008

Keywords

• base
• hard
• more
• nonn

Comments

Conjecture: the sequence is infinite.

The quotients are 1, 1, 3, 5, 6, 6, 7, 6, 7, 7, 7, 9, 9, 10, 10, 11.

From Nick Hobson, Feb 05 2024: (Start)

a(17) > 4500000.

Observation: the known terms of this sequence are near a power of 2:

k log_2(k)

1 0.000000

6 2.584963

122 6.930737

2126 11.053926

7910 12.949462

8254 13.010878

16201 13.983795

32312 14.979782

32426 14.984863

32998 15.010091

65436 15.997797

261649 17.997273

261803 17.998122

1044017 19.993714

1050183 20.002209

4194999 22.000239

Searching near 2^23, 2^24, and 2^25 finds term 16783381.

(End)

Examples

			6^6 = 1011011001000000_2; 1+0+1+1+0+1+1+0+0+1+0+0+0+0+0+0 = 6; 6 mod 6 = 0.
		

Links

• Nick Hobson, C program

Crossrefs

Cf. A108827.

Programs

C

See Links section.

Mathematica

Select[Range[1100000],Divisible[Total[IntegerDigits[#^#,2]],#]&] (* Harvey P. Dale, Dec 18 2014 *)

PARI

isok(k) = !(hammingweight(k^k) % k); \\ Michel Marcus, Aug 20 2021

Extensions

a(12)-a(15) from Lars Blomberg, Jul 01 2011

a(16) from Nick Hobson, Feb 05 2024