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

Author

Rémy Sigrist, Apr 29 2018

Keywords

• nonn
• tabl

Examples

			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
		

Links

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

Crossrefs

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

Programs

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)))

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).