A360597 -id:A360597 - cp's OEIS frontend

A360599 Ratios of consecutive terms of A360598: a(n) = max(A360598(n), A360598(n+1)) / min(A360598(n), A360598(n+1)).

1, 2, 3, 6, 4, 5, 20, 7, 8, 56, 9, 10, 90, 11, 12, 132, 13, 14, 182, 15, 16, 240, 17, 18, 306, 19, 21, 399, 22, 23, 506, 24, 25, 600, 26, 27, 702, 28, 29, 812, 30, 31, 930, 32, 33, 1056, 34, 35, 1190, 36, 37, 1332, 38, 39, 1482, 40, 41, 1640, 42, 43, 1806, 44

Offset: 1

Rémy Sigrist, Feb 13 2023

nonn

This sequence is a permutation of the positive integers with inverse A360600:
- we already know that all terms are distinct, so we just have to show that all integers appear,
- by contradiction: let r be the least value missing from this sequence,
- once the values 1..r-1 have appeared in this sequence, the sequence A360598 can only decrease finitely many times,
- the next increase in A360598 will correspond to the ratio r.

			For n = 15:
    A360598(15) = 11 and A360598(16) = 132,
    so a(15) = 132 / 11 = 12.
For n = 16:
    A360598(16) = 132 and A360598(17) = 1,
    so a(16) = 132 / 1 = 132.

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A360597, A360598, A360600 (inverse).

PARI
```
See Links section.
```

from itertools import islice
def agen(): # generator of terms
    an, ratios = 1, set()
    while True:
        k = 1
        q, r = divmod(max(k, an), min(k, an))
        while r != 0 or q in ratios:
            k += 1
            q, r = divmod(max(k, an), min(k, an))
        an = k
        ratios.add(q)
        yield q
print(list(islice(agen(), 66))) # Michael S. Branicky, Feb 13 2023

A371295 Value k at the n-th step of A371282, where multiplied values are positive and subtracted values are negative.

Original entry on oeis.org

2, 3, -1, 4, -17, 5, -11, 6, -16, 7, -49, 9, -54, 8, -62, 10, -89, 12, -120, 13, -143, 14, -168, 15, -194, 18, -271, 19, -305, 20, -341, 21, -378, 22, -440, 23, -483, 24, -527, 25, -599, 26, -649, 27, -701, 28, -755, 29, -811, 30, -869, 31, -929, 32, -991, 33

Offset: 1

Neal Gersh Tolunsky, Mar 17 2024

sign

			At the first step of A371282, to get from A371282(1)=1 to A371282(2)=2, we multiply by 2, so a(1)=2.
At the fifth step of A371282, to get from A371282(5)=20 to A371282(6)=3, we subtract 17, so a(5)=-17.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A371282, A099004 (steps of add or subtract), A360597 (steps of multiply or divide).

from itertools import islice
def agen(): # generator of terms
    mina, an, aset, mink, kset = 1, 1, {1}, 1, set()
    while True:
        k1, ak1, k2 = 0, mina, mink
        if mina < an:
            for ak1 in range(mina, an-mink+1):
                if ak1 not in aset and an - ak1 not in kset:
                    k1 = an - ak1
                    break
        while k2 in kset or an*k2 in aset:
            k2 += 1
        an, k = (an-k1, k1) if k1 > 0 else (an*k2, k2)
        yield -k if k1 > 0 else k
        aset.add(an)
        kset.add(k)
        while mina in aset: mina += 1
        while mink in kset: mink += 1
print(list(islice(agen(), 56))) # Michael S. Branicky, Mar 18 2024

a(12) and beyond from Michael S. Branicky, Mar 18 2024

cp's OEIS Frontend

A360599 Ratios of consecutive terms of A360598: a(n) = max(A360598(n), A360598(n+1)) / min(A360598(n), A360598(n+1)).

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