A309435 Iteratively replace the product of two sequentially chosen consecutive integers with those integers.

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

Links

Table of n, a(n) for n=1..70.

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

Adjacent sequences: A309432 A309433 A309434 * A309436 A309437 A309438

Keyword

nonn,easy

Author

Matthew Malone, Aug 02 2019

Status

approved