A302558 - OEIS

%I #16 Aug 20 2018 06:00:19

%S 0,0,1,0,3,7,5,0,1,9,18,28,30,23,13,0,15,31,48,66,73,64,50,33,11,29,5,

%T 29,54,55,29,0,31,63,41,16,51,87,124,162,201,241,252,231,207,180,150,

%U 117,73,113,152,192,233,275,318,362,364,321,275,226,174,119,61

%N For any n > 0 and m > 1, let d_m(n) be the distance from n to the nearest power of a number <= m (i.e., the distance to the nearest number of the form x^k with x <= m and k >= 0); a(n) = Sum_{i > 1} d_i(n).

%C For any n > 1 and m >= n, d_m(n) = 0, hence the series in the name contains only finitely many nonzero terms and is well defined.

%C The set of local minima (i.e., indices n > 1 where a(n) < min(a(n-1), a(n+1))) seem to correspond to A001597 minus {1, 9}.

%C See A303545 for a similar sequence.

%H Rémy Sigrist, <a href="/A302558/b302558.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A302558/a302558.png">Colored logarithmic pin plot of the first 1024 terms</a> (where the color is function of the number m in the term d_m(n))

%H <a href="/index/Di#distance_to_the_nearest">Index entries for sequences related to distance to nearest element of some set</a>

%F a(n) = 0 iff n is a power of 2.

%F a(n) >= A053646(n) (as d_2 = A053646).

%e For n = 10:

%e - d_2(10) = |10 - 8| = 2,

%e - d_m(10) = |10 - 9| = 1 for m = 3..9,

%e - d_m(10) = 0 for any m >= 10,

%e - hence a(10) = 2 + 7*1 = 9.

%o (PARI) a(n) = my (v=0, d=oo); for (m=2, oo, my (k=logint(n,m)); d = min(d, min(n-m^k, m^(k+1)-n)); if (d, v+=d, return (v)))

%Y Cf. A001597, A053646, A303545.

%K nonn

%O 1,5

%A _Rémy Sigrist_, Aug 15 2018