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A309435
Iteratively replace the product of two sequentially chosen consecutive integers with those integers.
1
1, 2, 3, 4, 5, 2, 3, 7, 8, 9, 5, 2, 11, 3, 4, 13, 14, 15, 16, 17, 18, 19, 4, 5, 3, 7, 2, 11, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 11, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 9, 5, 46, 47, 48, 49, 50, 51, 4, 13, 53, 54, 55, 7, 8, 57, 58, 59, 60
OFFSET
1,2
FORMULA
To generate the sequence, start with the integers, A_0={1,2,3,4,5,...}. To generate A_{n+1} calculate x = A_n(n) * A_n(n+1). Replace the next instance of x in A_n (after A_n(n+1)) with A_n(n), A_n(n+1). The limit of this process gives the sequence.
EXAMPLE
A_0 = {1,2,3,4,5,...}
A_0(0) * A_0(1) = 1 * 2 = 2, which is not found after A_0(1), so A_1 = A_0.
A_1(1) * A_1(2) = 2 * 3 = 6, which *is* found after A_1(2), so A_2 = {1,2,3,4,5,2,3,7,8,...}
A_2(2) * A_2(3) = 3 * 4 = 12, A_3 = {1,2,3,4,5,2,3,7,8,9,10,11,3,4,13,...}
A_3(3) * A_3(4) = 4 * 5 = 20, A_4 = {1,2,3,4,5,2,3,7,8,9,10,11,3,4,13,...,19,4,5,21,...}
A_4(4) * A_4(5) = 5 * 2 = 10, A_5 = {1,2,3,4,5,2,3,7,8,9,5,2,11,3,4,13,...,19,4,5,21,...}
MATHEMATICA
T = Range[100]; Do[p = T[[i]] T[[i + 1]]; Do[If[T[[j]] == p, T = Join[ T[[;; j-1]], {T[[i]], T[[i+1]]}, T[[j+1 ;; ]]]; Break[]], {j, i+2, Length@ T}], {i, Length@T}]; T (* Giovanni Resta, Sep 20 2019 *)
PROG
(Kotlin)
fun generate(len: Int): List<Int> {
fun gen_inner(len: Int, level: Int): List<Int> {
if (level < 1) return (1..len).toList()
val prev = gen_inner(len, level - 1)
if (level == len) return prev.take(len)
val (a, b) = prev[level - 1] to prev[level]
return if (prev.drop(level + 1).contains(a*b)) {
prev.indexOfFirst { it == a*b }.let { idx ->
prev.take(idx) + a + b + prev.drop(idx + 1)
}
} else prev
}
return gen_inner(len, len)
}
(PARI) a = vector(92, k, k); for (n=1, #a, print1 (a[n] ", "); s=a[n]*a[n+1]; for (k=n+2, #a, if (a[k]==s, a=concat([a[1..k-1], a[n..n+1], a[k+1..#a]]); break))) \\ Rémy Sigrist, Aug 03 2019
CROSSREFS
This sequence is similar to A309503 except it uses multiplication instead of addition.
Sequence in context: A213925 A141810 A141809 * A043265 A194459 A143120
KEYWORD
nonn,easy
AUTHOR
Matthew Malone, Aug 02 2019
STATUS
approved