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

0,2

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)

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 *)

Prog

(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

Crossrefs

Sequence in context: A213324 A007672 A322255 * A371359 A323615 A204934

Adjacent sequences: A084334 A084335 A084336 * A084338 A084339 A084340

Keyword

nonn

Author

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

Extensions

Corrected and extended by David Wasserman, Dec 15 2004

Status

approved