cp's OEIS Frontend

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

%I A138572 #16 Feb 05 2024 11:21:29
%S A138572 1,6,122,2126,7910,8254,16201,32312,32426,32998,65436,261649,261803,
%T A138572 1044017,1050183,4194999
%N A138572 Numbers k that divide the sum of the digits of k^k in base 2.
%C A138572 Conjecture: the sequence is infinite.
%C A138572 The quotients are 1, 1, 3, 5, 6, 6, 7, 6, 7, 7, 7, 9, 9, 10, 10, 11.
%C A138572 From _Nick Hobson_, Feb 05 2024: (Start)
%C A138572 a(17) > 4500000.
%C A138572 Observation: the known terms of this sequence are near a power of 2:
%C A138572        k    log_2(k)
%C A138572        1    0.000000
%C A138572        6    2.584963
%C A138572      122    6.930737
%C A138572     2126   11.053926
%C A138572     7910   12.949462
%C A138572     8254   13.010878
%C A138572    16201   13.983795
%C A138572    32312   14.979782
%C A138572    32426   14.984863
%C A138572    32998   15.010091
%C A138572    65436   15.997797
%C A138572   261649   17.997273
%C A138572   261803   17.998122
%C A138572  1044017   19.993714
%C A138572  1050183   20.002209
%C A138572  4194999   22.000239
%C A138572 Searching near 2^23, 2^24, and 2^25 finds term 16783381.
%C A138572 (End)
%H A138572 Nick Hobson, <a href="/A138572/a138572.c.txt">C program</a>
%e A138572 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 A138572 Select[Range[1100000],Divisible[Total[IntegerDigits[#^#,2]],#]&] (* _Harvey P. Dale_, Dec 18 2014 *)
%o A138572 (PARI) isok(k) = !(hammingweight(k^k) % k); \\ _Michel Marcus_, Aug 20 2021
%o A138572 (C) See Links section.
%Y A138572 Cf. A108827.
%K A138572 base,hard,more,nonn
%O A138572 1,2
%A A138572 _Robert Gerbicz_, May 12 2008
%E A138572 a(12)-a(15) from _Lars Blomberg_, Jul 01 2011
%E A138572 a(16) from _Nick Hobson_, Feb 05 2024