A303545 -id:A303545 - cp's OEIS frontend

A303548 For any n > 0 and h > 0, let d_h(n) be the distance from n to the nearest number with Hamming weight at most h; a(n) = Sum_{i > 0} d_i(n).

0, 0, 1, 0, 1, 2, 2, 0, 1, 2, 4, 4, 4, 4, 3, 0, 1, 2, 4, 4, 6, 8, 9, 8, 8, 8, 9, 8, 7, 6, 4, 0, 1, 2, 4, 4, 6, 8, 9, 8, 10, 12, 15, 16, 17, 18, 18, 16, 16, 16, 17, 16, 17, 18, 18, 16, 15, 14, 14, 12, 10, 8, 5, 0, 1, 2, 4, 4, 6, 8, 9, 8, 10, 12, 15, 16, 17, 18

Offset: 1

Rémy Sigrist, Apr 26 2018

For any n > 0 and h >= A000120(n), d_h(n) = 0, hence the series in the name contains only finitely many nonzero terms and is well defined.

See also A303545 for a similar sequence.

			For n = 42:
- d_1(n) = |42 - 32| = 10,
- d_2(n) = |42 - 40| = 2,
- d_h(n) = 0 for any h >= 3,
- hence a(42) = 10 + 2 = 12.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored pin plot of the first 3 * 512 terms (where the color is function of the number h in the term d_h(n))
Index entries for sequences related to distance to nearest element of some set

Cf. A000120, A053646, A303545.

a(n) = my (v=0, h=hamming weight(n)); for (d=0, oo, my (o=min(hamming weight(n-d), hamming weight(n+d))); if (o

a(n) = 0 iff n is a power of 2.
Apparently, a(2 * n) = 2 * a(n).
a(n) >= A053646(n) (as d_1 = A053646).

A302558 For any n > 0 and m > 1, let d_m(n) be the distance from n to the nearest power of a number <= m (i.e., the distance to the nearest number of the form x^k with x <= m and k >= 0); a(n) = Sum_{i > 1} d_i(n).

Original entry on oeis.org

0, 0, 1, 0, 3, 7, 5, 0, 1, 9, 18, 28, 30, 23, 13, 0, 15, 31, 48, 66, 73, 64, 50, 33, 11, 29, 5, 29, 54, 55, 29, 0, 31, 63, 41, 16, 51, 87, 124, 162, 201, 241, 252, 231, 207, 180, 150, 117, 73, 113, 152, 192, 233, 275, 318, 362, 364, 321, 275, 226, 174, 119, 61

Offset: 1

Rémy Sigrist, Aug 15 2018

nonn

For any n > 1 and m >= n, d_m(n) = 0, hence the series in the name contains only finitely many nonzero terms and is well defined.
The set of local minima (i.e., indices n > 1 where a(n) < min(a(n-1), a(n+1))) seem to correspond to A001597 minus {1, 9}.
See A303545 for a similar sequence.

			For n = 10:
- d_2(10) = |10 - 8| = 2,
- d_m(10) = |10 - 9| = 1 for m = 3..9,
- d_m(10) = 0 for any m >= 10,
- hence a(10) = 2 + 7*1 = 9.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic pin plot of the first 1024 terms (where the color is function of the number m in the term d_m(n))
Index entries for sequences related to distance to nearest element of some set

Cf. A001597, A053646, A303545.

a(n) = my (v=0, d=oo); for (m=2, oo, my (k=logint(n,m)); d = min(d, min(n-m^k, m^(k+1)-n)); if (d, v+=d, return (v)))

a(n) = 0 iff n is a power of 2.

a(n) >= A053646(n) (as d_2 = A053646).

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.

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

A303548 For any n > 0 and h > 0, let d_h(n) be the distance from n to the nearest number with Hamming weight at most h; a(n) = Sum_{i > 0} d_i(n).

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A302558 For any n > 0 and m > 1, let d_m(n) be the distance from n to the nearest power of a number <= m (i.e., the distance to the nearest number of the form x^k with x <= m and k >= 0); a(n) = Sum_{i > 1} d_i(n).

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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).

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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).

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Keywords

Examples

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Programs

PARI

Formula