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.

A384484 Smallest number with shortest addition-composition chain of length n, starting with 1 and x, i.e., smallest k such that A384483(k) = n.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 19, 70, 167, 1239, 7123
Offset: 0

Views

Author

Pontus von Brömssen, Jun 02 2025

Keywords

Comments

See A384480 and A384483 for details.

Crossrefs

Cf. A003064 (addition only), A383001 (addition and multiplication), A384385 (addition, multiplication, and composition), A384480, A384481, A384483, A384485.

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.

A384480 Square array read by antidiagonals: T(n,k) is the length of a shortest addition-composition chain for n*x+k, starting with 1 and x; n, k >= 0.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Jun 02 2025

Keywords

Comments

An addition-composition chain for the affine function f is a finite sequence of affine functions, starting with 1, x and ending with f, in which each element except 1 and x equals g(x)+h(x) or g(h(x)) for two preceding, not necessarily distinct, elements g(x) and h(x) in the chain. The length of the chain is the number of elements in the chain, excluding 1 and x. Such chains exist only for functions of the form f(x) = n*x+k, where n and k are nonnegative integers, not both 0.
T(0,0) = 0 by convention.
Equivalently, the chains can be defined on pairs (s,t) of nonnegative integers (corresponding to the function f(x) = s*x+t) with the operations (s,t)+(u,v) = (s+t,u+v) (addition) and (s,t)o(u,v) = (s*u,s*v+t) (composition).

Examples

			Array begins:
  n\k| 0  1  2  3  4  5  6  7  8  9 10 11 12
  ---+--------------------------------------
   0 | 0  0  1  2  2  3  3  4  3  4  4  5  4
   1 | 0  1  2  3  3  4  4  5  4  5  5  5  5
   2 | 1  2  2  3  3  4  4  4  4  5  5  5  5
   3 | 2  3  3  3  4  4  4  5  5  5  5  5  5
   4 | 2  3  3  3  3  4  3  4  4  4  4  5  4
   5 | 3  4  4  4  4  4  4  4  5  5  5  5  5
   6 | 3  4  4  4  4  4  4  5  4  4  5  5  5
   7 | 4  5  5  5  5  5  5  5  5  5  5  6  6
   8 | 3  4  4  4  4  4  4  4  4  5  4  5  4
   9 | 3  4  5  4  4  5  5  5  4  5  5  5  4
  10 | 4  5  5  5  5  5  5  5  5  5  5  6  5
  11 | 4  5  6  5  5  5  6  6  6  5  5  6  6
  12 | 4  5  5  5  5  5  5  5  5  5  5  5  5
For (n,k) = (4,6), the unique shortest chain for 4*x+6 is (1, x,) x+1, 2*x+2, 4*x+6 of length T(4,6) = 3. The last term of the chain is the composition of 2*x+2 with itself.
For (n,k) = (6,4), a shortest chain for 6*x+4 is (1, x,) x+1, 2*x+2, 3*x+2, 6*x+4 of length T(6,4) = 4. This chain uses only additions.

Crossrefs

Cf. A383330 (addition only), A384481, A384482, A384483 (row 0).

Formula

T(n,k) <= T(n,k-1) + 1.
T(n,k) <= T(n-1,k) + 1.

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.
Showing 1-4 of 4 results.

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