A138572 - OEIS

%I #16 Feb 05 2024 11:21:29

%S 1,6,122,2126,7910,8254,16201,32312,32426,32998,65436,261649,261803,

%T 1044017,1050183,4194999

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

%C Conjecture: the sequence is infinite.

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

%C From _Nick Hobson_, Feb 05 2024: (Start)

%C a(17) > 4500000.

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

%C        k    log_2(k)

%C        1    0.000000

%C        6    2.584963

%C      122    6.930737

%C     2126   11.053926

%C     7910   12.949462

%C     8254   13.010878

%C    16201   13.983795

%C    32312   14.979782

%C    32426   14.984863

%C    32998   15.010091

%C    65436   15.997797

%C   261649   17.997273

%C   261803   17.998122

%C  1044017   19.993714

%C  1050183   20.002209

%C  4194999   22.000239

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

%C (End)

%H Nick Hobson, <a href="/A138572/a138572.c.txt">C program</a>

%e 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.

%t Select[Range[1100000],Divisible[Total[IntegerDigits[#^#,2]],#]&] (* _Harvey P. Dale_, Dec 18 2014 *)

%o (PARI) isok(k) = !(hammingweight(k^k) % k); \\ _Michel Marcus_, Aug 20 2021

%o (C) See Links section.

%Y Cf. A108827.

%K base,hard,more,nonn

%O 1,2

%A _Robert Gerbicz_, May 12 2008

%E a(12)-a(15) from _Lars Blomberg_, Jul 01 2011

%E a(16) from _Nick Hobson_, Feb 05 2024