cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A317443 a(n) = number of k with 1 <= k <= n-1 such that a(k) + a(n-k) is not squarefree.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 7, 0, 7, 0, 10, 0, 9, 2, 8, 4, 9, 8, 10, 8, 11, 10, 17, 8, 13, 12, 15, 4, 19, 4, 25, 6, 17, 8, 25, 12, 19, 10, 27, 10, 23, 14, 22, 16, 25, 16, 26, 12, 21, 16, 22, 14, 31, 16, 40, 14, 29, 20, 32, 14, 27, 22, 35, 24, 33, 24, 32, 28, 25, 18
Offset: 1

Views

Author

Rémy Sigrist, Jul 28 2018

Keywords

Comments

We consider that 0 is not squarefree.
The scatterplot of the sequence has stripes linked to the 2-adic valuation of n.
See A317420 for similar sequences.

Examples

			For n = 4:
- a(1) + a(3) = 0 + 0 = 0 is not squarefree,
- a(2) + a(2) = 1 + 1 = 2 is squarefree,
- a(3) + a(1) = 0 + 0 = 0 is not squarefree,
- hence a(4) = 2.

Links

Crossrefs

Cf. A007814, A013929, A317420.

Programs

  • PARI
    a = vector(71); for (n=1, #a, a[n] = sum(k=1, n-1, !issquarefree(a[k]+a[n-k])); print1 (a[n] ", "))

A317582 a(n) is the number of k with 1 <= k <= n-1 such that a(k) * a(n-k) <= n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 6, 4, 4, 4, 6, 8, 8, 8, 8, 6, 7, 8, 9, 10, 9, 6, 6, 6, 8, 10, 14, 12, 12, 10, 8, 10, 12, 14, 14, 14, 8, 6, 10, 12, 18, 16, 14, 12, 9, 12, 15, 20, 21, 18, 16, 8, 12, 18, 20, 16, 16, 14, 14, 14, 14, 20, 23, 18, 16, 16, 14, 18, 22, 22, 22, 16
Offset: 1

Views

Author

Rémy Sigrist, Aug 01 2018

Keywords

Comments

This sequence can be described as a(n) = Sum_{k=1..n-1} [Q(a(k), a(n-k), n)] for some predicate Q in three variables, one of which corresponds to n; in that sense, this is a generalization of the sequences described in A317420.
See A317596 and A317638 for similar sequences.

Examples

			For n = 9:
- a(1) * a(8) = 0 * 6 = 0 <= 9,
- a(2) * a(7) = 1 * 6 = 6 <= 9,
- a(3) * a(6) = 2 * 5 = 10 > 9,
- a(4) * a(5) = 3 * 4 = 12 > 9,
- a(5) * a(4) = 4 * 3 = 12 > 9,
- a(6) * a(3) = 5 * 2 = 10 > 9,
- a(7) * a(2) = 6 * 1 = 6 <= 9,
- a(8) * a(1) = 6 * 0 = 0 <= 9,
- hence a(9) = 4.

Links

Crossrefs

Cf. A317420, A317596, A317638.

Programs

  • PARI
    a = vector(73); for (n=1, #a, a[n] = sum(k=1, n-1, a[k]*a[n-k] <= n); print1 (a[n] ", "))

A317419 a(n) = number of k with 1 <= k <= n-1 such that a(k) AND a(n-k) = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 6, 8, 8, 6, 6, 8, 8, 8, 10, 12, 12, 10, 14, 20, 14, 8, 10, 12, 12, 8, 8, 12, 14, 14, 10, 10, 12, 12, 8, 10, 14, 16, 12, 6, 10, 12, 14, 8, 8, 12, 14, 14, 14, 14, 10, 12, 12, 8, 12, 18, 16, 12, 8, 10, 14, 18, 14, 10, 18, 18, 16, 12, 14, 18
Offset: 1

Views

Author

Rémy Sigrist, Jul 27 2018

Keywords

Comments

All terms are even except a(2) = 1.
See A317420 for similar sequences.

Examples

			For n = 4:
- a(1) AND a(3) = 0 AND 2 = 0,
- a(2) AND a(2) = 1 AND 1 = 1 <> 0,
- a(3) AND a(1) = 2 AND 0 = 0,
- hence a(4) = 2.

Links

Crossrefs

Cf. A317420.

Programs

  • PARI
    a = vector(71); for (n=1, #a, a[n] = sum(k=1, n-1, bitand(a[k], a[n-k])==0); print1 (a[n] ", "))

A317441 a(n) = number of k with 1 <= k <= n-1 such that a(k) and a(n-k) have the same binary length (A070939).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 0, 5, 4, 1, 6, 3, 0, 5, 4, 1, 10, 5, 2, 7, 0, 7, 10, 5, 6, 7, 2, 5, 8, 5, 8, 7, 10, 9, 6, 7, 8, 7, 10, 11, 6, 7, 8, 11, 6, 17, 8, 9, 8, 19, 8, 13, 10, 15, 6, 19, 10, 9, 6, 19, 8, 13, 14, 13, 10, 17, 14, 15, 10, 19, 14, 9, 16, 15, 10, 11, 20
Offset: 1

Views

Author

Rémy Sigrist, Jul 28 2018

Keywords

Comments

See A317420 for similar sequences.

Examples

			For n = 4:
- A070939(a(1)) = 1 <> 2 = A070939(a(3)),
- A070939(a(2)) = 1 =  1 = A070939(a(2)),
- A070939(a(3)) = 2 <> 1 = A070939(a(1)),
- hence a(4) = 1.

Links

Crossrefs

Cf. A070939, A317420.

Programs

  • PARI
    l = vector(77); for (n=1, #l, l[n] = #binary(max(1, v=sum(k=1, n-1, l[k]==l[n-k]))); print1 (v ", "))

A317585 a(n) = number of k with 1 <= k <= n-1 such that abs(a(k) - a(n-k)) <= 2.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 4, 3, 2, 9, 6, 7, 6, 5, 2, 5, 6, 9, 4, 9, 2, 13, 12, 13, 8, 7, 6, 11, 6, 13, 2, 9, 8, 7, 8, 13, 4, 13, 6, 15, 6, 11, 12, 13, 16, 23, 8, 13, 0, 17, 14, 25, 14, 13, 8, 7, 6, 19, 8, 19, 6, 13, 4, 19, 12, 25, 20, 23, 10, 19, 10, 21, 16, 23, 10
Offset: 1

Views

Author

Rémy Sigrist, Aug 01 2018

Keywords

Comments

See A317420 for similar sequences.

Links

Crossrefs

Cf. A317420.

Programs

  • PARI
    a = vector(75); for (n=1, #a, a[n] = sum(k=1, n-1, abs(a[k]-a[n-k])<=2); print1 (a[n] ", "))

Formula

For n = 5:
- |a(1) - a(4)| = |0 - 3| = 3 > 2,
- |a(2) - a(3)| = |1 - 2| = 1 <= 2,
- |a(3) - a(2)| = |2 - 1| = 1 <= 2,
- |a(4) - a(1)| = |3 - 0| = 3 > 2,
- hence a(5) = 2.

A317922 a(n) = number of k with 0 < 2*k < n-1 such that a(n-k) AND a(n-2*k) = a(n-k) (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 0, 2, 1, 2, 1, 3, 3, 4, 0, 3, 2, 4, 2, 4, 3, 5, 1, 3, 3, 7, 2, 4, 4, 7, 0, 4, 3, 6, 5, 7, 4, 8, 2, 6, 4, 12, 1, 10, 5, 7, 2, 7, 1, 9, 3, 5, 6, 9, 4, 7, 3, 7, 3, 11, 5, 8, 3, 8, 4, 10, 3, 11, 6, 11, 1, 9, 4, 11, 8, 10, 8, 13, 2, 11, 7, 15
Offset: 1

Views

Author

Rémy Sigrist, Aug 11 2018

Keywords

Comments

This sequence has similarities with A317420.

Examples

			For n = 5:
- a(5-1) AND a(5-2) = 0 AND 1 = 0 = a(5-1),
- a(5-2) AND a(5-4) = 1 AND 0 = 0 <> a(5-2),
- hence a(5) = 1.

Links

Crossrefs

Cf. A000265, A001511, A317420.
Showing 1-6 of 6 results.

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