A302558 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).

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, 29, 54, 55, 29, 0, 31, 63, 41, 16, 51, 87, 124, 162, 201, 241, 252, 231, 207, 180, 150, 117, 73, 113, 152, 192, 233, 275, 318, 362, 364, 321, 275, 226, 174, 119, 61

Offset

1,5

Comments

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.

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}.

See A303545 for a similar sequence.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Colored logarithmic pin plot of the first 1024 terms (where the color is function of the number m in the term d_m(n))

Index entries for sequences related to distance to nearest element of some set

Formula

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

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

Example

For n = 10:
- d_2(10) = |10 - 8| = 2,
- d_m(10) = |10 - 9| = 1 for m = 3..9,
- d_m(10) = 0 for any m >= 10,
- hence a(10) = 2 + 7*1 = 9.

Prog

(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)))

Crossrefs

Cf. A001597, A053646, A303545.

Sequence in context: A291534 A331733 A301755 * A193506 A086242 A322931

Adjacent sequences: A302555 A302556 A302557 * A302559 A302560 A302561

Keyword

nonn

Author

Rémy Sigrist, Aug 15 2018

Status

approved