A303548 -id:A303548 - 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).

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).

Original entry on oeis.org

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

Rémy Sigrist, Apr 29 2018

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.

			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.

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

Cf. A002034, A303545, A303548, A303703.

PARI
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See Links section.
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a(n) = 0 iff n belongs to A303703.

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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