A303545 For any n > 0 and prime number p, let d_p(n) be the distance from n to the nearest p-smooth number; a(n) = Sum_{i prime} d_i(n).

0, 0, 1, 0, 2, 2, 3, 0, 1, 3, 6, 4, 7, 5, 2, 0, 6, 2, 9, 6, 9, 11, 14, 8, 8, 10, 5, 6, 12, 4, 10, 0, 4, 9, 5, 4, 15, 16, 13, 12, 24, 18, 28, 18, 16, 22, 28, 16, 17, 16, 20, 20, 25, 10, 12, 12, 17, 22, 24, 8, 21, 13, 3, 0, 5, 8, 26, 18, 16, 10, 25, 8, 28, 21

Offset

1,5

Comments

For any n > 0 and prime number p >= A006530(n), d_p(n) = 0; hence the series in the name contains only finitely many nonzero terms and is well defined.

See also A303548 for a similar sequence.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..32768

Rémy Sigrist, Colored pin plot of the first 3 * 512 terms (where the color is function of the prime p in the term d_p(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(2 * n) <= 2 * a(n).

a(n) >= A053646(n) + A301574(n) (as d_2 = A053646 and d_3 = A301574).

Example

For n = 42:
- d_2(42) = |42 - 32| = 10,
- d_3(42) = |42 - 36| = |42 - 48| = 6,
- d_5(42) = |42 - 40| = 2,
- d_p(42) = 0 for any prime number p >= 7,
- hence a(42) = 10 + 6 + 2 = 18.

Prog

(PARI) gpf(n) = if (n==1, 1, my (f=factor(n)); f[#f~, 1])
a(n) = my (v=0, pi=primepi(gpf(n))); for (d=0, oo, my (o=min(primepi(gpf(n-d)), primepi(gpf(n+d)))); if (o<pi, v += d*(pi-o); pi=o); if (o<=1, return (v)))

Crossrefs

Cf. A006530, A053646, A301574, A303548.

Sequence in context: A257398 A182631 A231728 * A306191 A290125 A307356

Adjacent sequences: A303542 A303543 A303544 * A303546 A303547 A303548

Keyword

nonn,look

Author

Rémy Sigrist, Apr 26 2018

Status

approved