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).

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

1,16

Links

Table of n, a(n) for n=1..87.

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

Formula

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).

Example

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

Prog

(PARI) 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)))

Crossrefs

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

Sequence in context: A057764 A010108 A033782 * A321740 A098082 A321921

Adjacent sequences: A302718 A302719 A302720 * A302722 A302723 A302724

Keyword

nonn,tabl

Author

Rémy Sigrist, Apr 29 2018

Status

approved