A384484 -id:A384484 - cp's OEIS frontend

A384483 Length of shortest addition-composition chain for n, starting with 1 and x.

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

Offset: 1

Pontus von Brömssen, Jun 02 2025

nonn

See A384480 for the definition of addition-composition chains. The number n is identified with the constant function f(x) = n.

			The smallest n for which a(n) < A003313(n) is n = 21. The length of a shortest addition chain for 21 is A003313(21) = 6, but there are addition-composition chains of length 5, for example (1, x,) x+1, 2*x+2, 3*x+3, 6, 21. 6 and 21 are the compositions of 3*x+3 with 1 and 6, respectively.

Row 0 of A384480 for columns k >= 1.

Cf. A003313 (addition only), A230697 (addition and multiplication), A384384 (addition, multiplication, and composition), A384484, A384485.

a(n) <= A003313(n).

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

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

Pontus von Brömssen, Jun 02 2025

See A384480 and A384483 for details.

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

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

Pontus von Brömssen, Jun 01 2025

See A384383 and A384384 for details.

Cf. A383001 (addition and multiplication), A384383, A384384, A384386, A384484 (addition and composition).

A384481 Smallest value of f(1) for a function f(x) = b*x+c with nonnegative integer coefficients and a shortest addition-composition chain of length n, starting with 1 and x.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 13, 24, 46, 98

Offset: 0

Pontus von Brömssen, Jun 02 2025

See A384480 for details.

			  n | functions b*x+c with b+c = a(n) and shortest chain of length n
  --+---------------------------------------------------------------
  0 | 1, x
  1 | 2, x+1, 2*x
  2 | 3, x+2, 2*x+1, 3*x
  3 | 3+x, x+3
  4 | 5+x, x+5
  5 | 7+x, x+7
  6 | 11*x+2
  7 | 23*x+1
  8 | 7*x+39, 43*x+3
  9 | 11*x+87

Cf. A383332 (addition only), A384480, A384482, A384484.

cp's OEIS Frontend

A384483 Length of shortest addition-composition chain for n, starting with 1 and x.

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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.

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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.

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Author

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A384481 Smallest value of f(1) for a function f(x) = b*x+c with nonnegative integer coefficients and a shortest addition-composition chain of length n, starting with 1 and x.

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