A084337 Rearrangement of positive integers so that the successive ratios (of the larger to the smaller term) are all distinct integers. a(m)/a(m-1) = a(k)/a(k-1) iff m = k (assuming a(m) > a(m-1), otherwise the ratio a(m-1)/a(m) is to be considered). Priority is given to smallest number not included earlier rather than to the successive ratio that has not occurred earlier.

1, 2, 6, 24, 3, 15, 90, 5, 35, 315, 7, 70, 770, 10, 120, 4, 52, 728, 8, 128, 1920, 12, 204, 3876, 17, 340, 7140, 14, 308, 11, 253, 6072, 22, 550, 14300, 13, 351, 9, 261, 8091, 29, 928, 16, 528, 17952, 32, 1120, 20, 720, 18, 666, 25308, 19, 779, 32718, 21, 903, 39732

Offset: 0

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 18 2003

Keywords

• nonn

Comments

The sequence of successive ratios is 2/1, 6/2, 24/6, 24/3, 15/3, 90/15, 90/9, 63/9, 63/7, ... or 2, 3, 4, 8, 5, 6, 10, 7, 9, ...

Links

• Michael S. Branicky, Table of n, a(n) for n = 0..10000 (0..5000 from Ivan Neretin)

Programs

Mathematica

a = r = {1}; Do[If[(ds = Select[Divisors[a[[-1]]], ! MemberQ[a, #] && ! MemberQ[r, a[[-1]]/#] &, 1]) != {}, nxta = ds[[1]]; nxtr = a[[-1]]/nxta, k = 1; While[MemberQ[r, k] || MemberQ[a, a[[-1]]*k], k++]; nxtr = k; nxta = k*a[[-1]]]; AppendTo[a, nxta]; AppendTo[r, nxtr], {n, 57}]; a (* Ivan Neretin, Jul 05 2015 *)

Python

from sympy import divisors
from itertools import islice
def agen(): # generator of terms
    mina, an, aset, mink, kset = 1, 1, {1}, 1, set()
    while True:
        yield an
        k1, ak1, k2 = 0, mina, mink
        if mina < an:
            for d in divisors(an):
                if d not in aset and an//d not in kset:
                    k1 = an//d
                    break
        while k2 in kset or an*k2 in aset:
            k2 += 1
        an, k = (an//k1, k1) if k1 > 0 else (an*k2, k2)
        aset.add(an)
        kset.add(k)
        while mina in aset: mina += 1
        while mink in kset: mink += 1
print(list(islice(agen(), 58))) # Michael S. Branicky, Mar 18 2024

Extensions

Corrected and extended by David Wasserman, Dec 15 2004