cp's OEIS Frontend

A292788 For n > 1, a(n) = least positive k, not a power of n, such that the digital sum of k in base n equals the digital sum of k^3 in base n.

%I A292788 #19 Sep 26 2017 08:24:03
%S A292788 56953,13,2,3,20,2,6,3,8,5,1110,3,65,8,4,7,86,9,2374,4,8,12,114,3,99,
%T A292788 12,135,15,3567,4,185,15,11,16,6,19,73,20,12,5,81,6,85,23,19,24,93,7,
%U A292788 97,24,18,27,796,28,44,7,19,28,413,4,365,32,8,31,26,21,200
%N A292788 For n > 1, a(n) = least positive k, not a power of n, such that the digital sum of k in base n equals the digital sum of k^3 in base n.
%C A292788 The term a(10) = 8 belongs to A070276.
%C A292788 For any n > 1, a(n^2) <= n.
%C A292788 Is this sequence defined for any n > 1 ?
%C A292788 Apparently, a(k) < k for any odd k > 3.
%C A292788 Among the first 99 999 terms, the digital sum of a(n) in base n is > n for n = 2, 12, 20, 30.
%C A292788 The scatterplot of the sequence shows beams on the upper part, which correspond to clusters of close points for which a(n) = k*n + (n-k-e) for some k > 0 and e in { 0, 2 }.
%C A292788 See also A292787 for a similar sequence involving squares instead of cubes.
%C A292788 The least positive k, not a power of 2, such that the hamming weight of k equals the hamming weight of k^4 is 34225258495.
%H A292788 Rémy Sigrist, <a href="/A292788/b292788.txt">Table of n, a(n) for n = 2..10000</a>
%H A292788 Rémy Sigrist, <a href="/A292788/a292788.png">Colorized scatterplot of the sequence for n=2..100000</a>
%e A292788 For n = 3:
%e A292788 - let d_3 denote the digital sum in base 3 (d_3 = A053735),
%e A292788 - 1 is a power of 3,
%e A292788 - d_3(2) = 2 and d_3(2^3) = 4,
%e A292788 - 3 is a power of 3,
%e A292788 - d_3(4) = 2 and d_3(4^3) = 4,
%e A292788 - d_3(5) = 3 and d_3(5^3) = 7,
%e A292788 - d_3(6) = 2 and d_3(6^3) = 4,
%e A292788 - d_3(7) = 3 and d_3(7^3) = 5,
%e A292788 - d_3(8) = 4 and d_3(8^3) = 8,
%e A292788 - 9 is a power of 3,
%e A292788 - d_3(10) = 2 and d_3(10^3) = 4,
%e A292788 - d_3(11) = 3 and d_3(11^3) = 9,
%e A292788 - d_3(12) = 2 and d_3(12^3) = 4,
%e A292788 - d_3(13) = 3 and d_3(13^3) = 3,
%e A292788 - hence a(3) = 13.
%t A292788 With[{kk = 10^5}, Table[SelectFirst[Complement[Range[2, kk], n^Range@ Floor@ Log[n, kk]], Total@ IntegerDigits[#, n] == Total@ IntegerDigits[#^3, n] &] /. k_ /; MissingQ@ k -> -1, {n, 2, 68}]] (* _Michael De Vlieger_, Sep 24 2017 *)
%o A292788 (PARI) a(n) = my (p=1); for (k=1, oo, if (k==p, p*=n, if (sumdigits(k,n) == sumdigits(k^3,n), return (k))))
%Y A292788 Cf. A053735, A070276, A292787.
%K A292788 nonn,base,look
%O A292788 2,1
%A A292788 _Rémy Sigrist_, Sep 23 2017