A301574 -id:A301574 - cp's OEIS frontend

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

Rémy Sigrist, Apr 26 2018

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.

			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.

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

Cf. A006530, A053646, A301574, A303548.

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

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).

A302721 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the distance from n to the nearest prime(k)-smooth number (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Rémy Sigrist, Apr 29 2018

			Array T(n, k) begins:
  n\k|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  ---+------------------------------------------------------------
    1|  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    2|  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    3|  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    4|  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    5|  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    6|  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    7|  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    8|  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    9|  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   10|  2  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   11|  3  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   12|  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   13|  3  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0

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

Cf. A053646 (first column), A061395, A301574 (second column), A303545 (row sums).

gpf(n) = if (n==1, 1, my (f=factor(n)); f[#f~, 1])
T(n,k) = my (p=prime(k)); for (d=0, oo, if (gpf(n-d) <= p || gpf(n+d) <= p, return (d)))

a(2^i, k) = 0 for any i >= 0.
a(2*n, k) <= 2*a(n, k).
a(n, k+1) <= a(n, k).
abs(T(n+1, k) - T(n, k)) <= 1.
a(n, A061395(n)) = 0 for any n > 1.
a(n, 1) = A053646(n).
a(n, 2) = A301574(n).
Sum_{k > 0} a(n, k) = A303545(n).

cp's OEIS Frontend

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).

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