cp's OEIS Frontend

A287076 a(n) = least k > n with the same sum of digits as n in some base b > 1.

%I A287076 #14 May 21 2017 07:54:25
%S A287076 2,4,5,6,6,9,10,12,10,12,13,16,14,16,19,20,18,20,21,22,22,24,25,30,26,
%T A287076 28,29,30,30,36,33,34,34,36,37,40,38,40,42,42,42,44,45,48,46,48,49,56,
%U A287076 50,52,53,56,54,57,57,58,58,60,61,64,62,66,67,66,66,68,69
%N A287076 a(n) = least k > n with the same sum of digits as n in some base b > 1.
%C A287076 More formally: a(n) = Min_{b>1} f_b(n), where f_b(n) = least k > n with the same sum of digits as n in base b.
%C A287076 We have the following properties:
%C A287076 - f_b(b) = b^2 for any b > 1,
%C A287076 - f_b(b^k) = b^(k+1) for any b > 1 and k >= 0,
%C A287076 - f_b(n) = b + n - 1 for any b > 1 and n < b,
%C A287076 - f_b(n) - n >= b - 1 for any b > 1 and n > 0.
%C A287076 Also, f_2 = A057168 and f_10 = A228915.
%C A287076 For any n > 0, n < a(n) <= 2*n.
%C A287076 Conjecturally, a(n) ~ n.
%C A287076 The derived sequence e(n) = a(n) - n is unbounded: for any n > 0:
%C A287076 - for any b such that 1 < b <= n, let x_b = the least power of b such that f_b(i*x_b) - i*x_b >= n for any i > 0,
%C A287076 - let X = Lcm_{b=2..n} x_b,
%C A287076 - then f_b(X) - X >= n for any b such that 1 < b <= n,
%C A287076 - also, f_b(X) - X >= b - 1 >= n for any b > n,
%C A287076 - hence a(X) - X = e(X) >= n, QED.
%H A287076 Rémy Sigrist, <a href="/A287076/b287076.txt">Table of n, a(n) for n = 1..10000</a>
%H A287076 Rémy Sigrist, <a href="/A287076/a287076.gp.txt">PARI program for A287076</a>
%e A287076 The following table shows f_b(8) for all bases b > 1:
%e A287076 b    f_b(8)   8 in base b   f_b(8) in base b
%e A287076 --   ------   -----------   ----------------
%e A287076 2        16        "1000"            "10000"
%e A287076 3        14          "22"              "112"
%e A287076 4        17          "20"              "101"
%e A287076 5        12          "13"               "22"
%e A287076 6        13          "12"               "21"
%e A287076 7        14          "11"               "20"
%e A287076 8        64          "10"              "100"
%e A287076 b>8     b+7           "8"               "17"
%e A287076 Hence, a(8) = f_5(8) = 12.
%Y A287076 Cf. A057168, A228915.
%K A287076 nonn,base
%O A287076 1,1
%A A287076 _Rémy Sigrist_, May 19 2017