A303711 For any n > 0 and f > 0, let d_f(n) be the distance from n to the nearest number congruent mod f! to some divisor of f!; a(n) = Sum_{i > 0} d_i(n).

0, 0, 0, 1, 2, 0, 3, 0, 2, 3, 8, 0, 9, 4, 3, 6, 19, 8, 19, 4, 5, 11, 20, 0, 5, 12, 8, 5, 26, 0, 27, 6, 12, 19, 8, 4, 35, 24, 11, 5, 42, 10, 46, 13, 8, 26, 50, 8, 17, 18, 28, 29, 64, 16, 15, 8, 19, 41, 56, 0, 57, 30, 9, 14, 23, 27, 85, 36, 31, 15, 78, 12, 80

Offset: 1

Author

Rémy Sigrist, Apr 29 2018

Keywords

• nonn
• look

Comments

For any n > 0 and f >= A002034(n), d_f(n) = 0; hence the series in the name contains only finitely many nonzero terms and is well defined.

See also A303545 and A303548 for similar sequences; unlike these sequences, the indexed family {d_i, i > 0} used here does not satisfy for any n > 0 and f < g the inequality d_f(n) >= d_g(n); also d_i is i!-periodic for any i > 0.

Examples

			For n = 42:
- d_1(n) = 0,
- d_2(n) = 0,
- d_3(n) = 0,
- d_4(n) = |42 - 36| = |42 - 48| = 6,
- d_5(n) = |42 - 40| = 2,
- d_6(n) = |42 - 40| = 2,
- d_f(n) = 0 for any f >= 7,
- hence a(42) = 6 + 2 + 2 = 10.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored pin plot of the first 2000 terms (where the color is function of the number f in the term d_f(n))
• Rémy Sigrist, PARI program for A303711
• Index entries for sequences related to distance to nearest element of some set

Crossrefs

Cf. A002034, A303545, A303548, A303703.

Programs

PARI

See Links section.

Formula

a(n) = 0 iff n belongs to A303703.