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 1-4 of 4 results.

A384383 Number of polynomials with a shortest addition-multiplication-composition chain of length n, starting with 1 and x.

Original entry on oeis.org

2, 4, 14, 73, 586, 7250
Offset: 0

Views

Author

Pontus von Brömssen, Jun 01 2025

Keywords

Comments

An addition-multiplication-composition chain for the polynomial p(x) is a finite sequence of polynomials, starting with 1, x and ending with p(x), in which each element except 1 and x equals q(x)+r(x), q(x)*r(x), or q(r(x)) for two preceding, not necessarily distinct, elements q(x) and r(x) in the chain. The length of the chain is the number of elements in the chain, excluding 1 and x.

Examples

			An example of a polynomial for which composition is necessary to obtain the shortest chain is 9*x, with the chain (1, x,) 2*x, 3*x, 9*x. (9*x is the composition of 3*x with itself.) So 9*x is one of the 11 polynomials counted by a(3) but not by A384382(3).

Crossrefs

Cf. A382928, A383331 (addition only), A384382 (addition and multiplication), A384386, A384482 (addition and composition).

A384485 Number of integers with a shortest addition-composition chain of length n, starting with 1 and x, i.e., number of integers k with A384483(k) = n.

Original entry on oeis.org

1, 1, 2, 3, 5, 20, 104, 700, 6779, 95596
Offset: 0

Views

Author

Pontus von Brömssen, Jun 02 2025

Keywords

Comments

See A384480 and A384483 for details.

Crossrefs

Cf. A003065 (addition only), A383002 (addition and multiplication), A384386 (addition, multiplication, and composition), A384480, A384482, A384483, A384484.

A384384 Length of shortest addition-multiplication-composition chain for n, starting with 1 and x.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 4, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 6, 6, 5, 5, 6, 5, 5, 5, 6, 6, 6, 5, 6, 5, 6, 6, 6, 5, 6, 6, 6, 4, 5, 5, 6, 5, 6, 6, 6, 5, 6, 6, 5, 6, 6, 6, 6, 5, 4, 5, 5, 5, 6, 6, 6, 6, 6, 5, 5, 6, 6, 6, 7
Offset: 1

Views

Author

Pontus von Brömssen, Jun 01 2025

Keywords

Comments

See A384383 for the definition of addition-multiplication-composition chains. The number n is identified with the constant polynomial p(x) = n.

Examples

			The smallest n for which a(n) < A230697(n) is n = 31. The length of a shortest addition-multiplication chain for 31 is A230697(31) = 6, but there are addition-multiplication-composition chains of length 5, for example (1, x,) 2*x, 2*x+1, 4*x+3, 7, 31. (4*x+3 is the composition of 2*x+1 with itself; 7 and 31 are the compositions of 4*x+3 with 1 and 7, respectively.)

Crossrefs

Cf. A230697 (addition and multiplication), A384383, A384385, A384386, A384483 (addition and composition).

Formula

a(n) <= A230697(n).
a(n) <= a(n-1) + 1.

A384385 Smallest number with shortest addition-multiplication-composition chain of length n, starting with 1 and x, i.e., smallest k such that A384384(k) = n.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 23, 95
Offset: 0

Views

Author

Pontus von Brömssen, Jun 01 2025

Keywords

Comments

See A384383 and A384384 for details.

Crossrefs

Cf. A383001 (addition and multiplication), A384383, A384384, A384386, A384484 (addition and composition).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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