A386483 -id:A386483 - cp's OEIS frontend

A386482 a(1)=1, a(2)=2; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

1, 2, 4, 6, 3, 9, 12, 10, 8, 14, 7, 21, 18, 16, 20, 15, 5, 25, 30, 28, 26, 24, 22, 11, 33, 27, 36, 34, 32, 38, 19, 57, 54, 52, 50, 48, 46, 44, 42, 40, 35, 45, 39, 13, 65, 60, 58, 56, 49, 63, 51, 17, 68, 66, 64, 62, 31, 93, 90, 88, 86, 84, 82, 80, 78, 76, 74, 72, 70, 55, 75, 69, 23, 92, 94, 47, 141, 138, 136, 134, 132, 130, 128

Offset: 1

N. J. A. Sloane, Aug 15 2025, based on email messages from Geoffrey Caveney

Similar to the EKG sequence A064413, but whereas in that sequence a(n) is chosen to be as small as possible, here the primary goal is to choose a(n) to be less than a(n-1) and as close to it as possible. This sequence first differs from the EKG sequence at n = 8, where a(8) = k = 10 is closer to a(7) = 12 than A064413(8) = 8 is.
A significant difference from the EKG sequence is that the primes do not appear in their natural order. Also, it is not always true that a prime p is preceded by 2*p when it first appears. 4k+3 primes appear to be preceded by smaller multiples than 4k+1 primes.
It is conjectured that every positive number appears.
It is interesting to study what happens if the first two terms are taken to be 1,s, with s >= 2, or if the first s terms are taken to be 1,2,3,...,s, with s >= 2. Call two such sequences equivalent if they eventually merge. The 1,3 and 1,2,3 sequences merge with each other after half-a-dozen terms. But at present we do not know if they merge with the 1,2 sequence.
It appears that many sequences that start 1,s and 1,2,3,...,s with small s merge with one of the sequences 1,2 or 1,2,3 or 1,2,3,...,11.
[The preceding comments are from Geoffrey Caveney's emails.]
From Michael De Vlieger, Aug 15 2025: (Start)
There are long runs of terms with the same parity in this sequence. For example, beginning at a(481) = 948, there are 100 consecutive even terms. Starting with a(730076) = 1026330, there are 100869 consecutive even terms, followed by 36709 consecutive odd terms. Runs of even terms tend to be longer than those of odd.
There are long runs of first differences of -2 and -6 in this sequence, and that there appear to be three phases. The predominant (A) phase has a(n) = a(n-1)-2, the second (B) phase has a(n) = a(n-1)-6, and then there is a turbulent (C) phase [C] with varied differences.
Generally the even runs correspond to differences a(n)-a(n-1) = 2 and feature square-free terms separated by an odd number of terms in A126706. Phase [C] tends to be largely odd squarefree semiprimes and includes prime powers. (End)

Geoffrey Caveney, Emails to N. J. A. Sloane, Aug 13 2025 - Aug 15 2025.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers that are neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers.

Cf. A064413 (EKG), A387072 (inverse), A387073 (record high points), A387074 (indices of record high points), A387075 (first differences), A387076 (primes in order of appearance), A387077 (indices of primes), A387078 (run lengths of consecutive odd and even terms), A387080 (variant that begins with 1,3).

See also A386483, A386484, A387087-A387090.

aList[n_] := Module[{an = 2, aset = <|2 -> True|>, m}, Reap[Sow[1]; Sow[an];
Do[m = SelectFirst[Range[an - 1, 2, -1], ! KeyExistsQ[aset, #] && GCD[#, an] > 1 & ];
If[MissingQ[m], m = NestWhile[# + 1 &, an + 1, !(! KeyExistsQ[aset, #] && GCD[#, an] > 1) & ]];
aset[m] = True; an = m; Sow[an], {n - 2}]][[2, 1]]]; aList[83]  (* Peter Luschny, Aug 15 2025 *)

PARI
```
\\ See Links section.
```

from math import gcd
from itertools import count, islice
def A386482_gen(): # generator of terms
    yield 1
    an, aset = 2, {2}
    while True:
        yield an
        m = next((k for k in range(an-1, 1, -1) if k not in aset and gcd(k, an) > 1), False)
        if not m: m = next(k for k in count(an+1) if k not in aset and gcd(k, an) > 1)
        an = m
        aset.add(an)
print(list(islice(A386482_gen(), 83))) # Michael S. Branicky, Aug 15 2025

A386484 a(n) is the index where A387090(n) appears in A386482.

Original entry on oeis.org

1, 2, 5, 17, 24, 44, 52, 73, 126, 809, 821, 1016, 2020, 2026, 2637, 3017, 3541, 3544, 4876, 12680, 12803, 12808, 12898, 12901, 12907, 13279, 17729, 17734, 27224, 33351, 35295, 35310, 44605, 44609, 44925, 44930, 44935, 44939, 45176, 45279, 45703, 45723, 46125, 46138, 46154, 46157, 62957, 63713, 63720, 63869, 63872, 63877, 115110, 115116

Offset: 1

N. J. A. Sloane, Aug 16 2025

A387090 lists the numbers that are the slowest to appear in A386482, and the present sequence tells when they do appear. This is of interest because we do not know if every number appears in A386482.

Rémy Sigrist, Table of n, a(n) for n = 1..1243 (first 127 terms from N. J. A. Sloane)
Rémy Sigrist, C++ program

Cf. A386482, A386483, A387072, A387090.

A387522 Index of first term in A386482 that is divisible by the n-th prime.

Original entry on oeis.org

2, 4, 8, 10, 23, 21, 28, 30, 37, 47, 56, 67, 63, 61, 75, 94, 88, 86, 80, 119, 117, 135, 131, 174, 166, 162, 160, 156, 154, 150, 200, 235, 229, 227, 217, 215, 209, 270, 266, 260, 254, 252, 242, 240, 297, 295, 314, 354, 350, 348, 344, 338, 336, 326, 428, 422, 416, 414, 408, 404, 402, 392, 378, 374, 372, 478, 464, 458, 608, 606, 602

Offset: 1

N. J. A. Sloane, Sep 01 2025.

Michael De Vlieger, Table of n, a(n) for n = 1..65536 (first 841 terms from N. J. A. Sloane)

Cf. A386482, A386483, A387072, A387076.

Block[{p, s}, p[A386482/b386482.txt%22,%20%22Data%22%5D%5B%5BAll,%20-1%5D%5D;%20Do%5BMap%5BIf%5Bp%5B%23%5D%20==%200,%20Set%5Bp%5B%23%5D,%20n%5D%5D%20&,%20FactorInteger%5Bs%5B%5Bn%5D%5D%5D%5B%5B;;%20,%201%5D%5D%5D,%20%7Bn,%20Length%5Bs%5D%7D%5D;%20TakeWhile%5BArray%5Bp%5BPrime%5B%23%5D%5D%20&,%20120%5D,%20%23%20%3E%200%20&%5D%20%5D%20(*%20_Michael%20De%20Vlieger">] := 0; s = Import["https://oeis.org/A386482/b386482.txt", "Data"][[All, -1]]; Do[Map[If[p[#] == 0, Set[p[#], n]] &, FactorInteger[s[[n]]][[;; , 1]]], {n, Length[s]}]; TakeWhile[Array[p[Prime[#]] &, 120], # > 0 &] ] (* _Michael De Vlieger, Sep 03 2025 *)

A387523 Primes in the order in which they first divide a term of A386482.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 17, 19, 23, 29, 31, 43, 41, 37, 47, 67, 61, 59, 53, 73, 71, 83, 79, 113, 109, 107, 103, 101, 97, 89, 127, 157, 151, 149, 139, 137, 131, 193, 191, 181, 179, 173, 167, 163, 199, 197, 211, 251, 241, 239, 233, 229, 227, 223, 313, 311, 307, 293, 283, 281, 277, 271, 269, 263, 257, 337, 331, 317, 467, 463, 461, 457

Offset: 1

N. J. A. Sloane, Sep 03 2025

Conjecture: if c is the first term in A386482 that is divisible by a prime p > 2, then c = 2*p.

			11 first divides A386482(23), but 13 first divides A386482(21), so 13 precedes 11 in this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..65536 (first 841 terms from N. J. A. Sloane.)

Cf. A386482, A386483, A387072, A387076, A387077, A387522, A387524.

Block[{p, s}, p[A386482/b386482.txt%22,%20%22Data%22%5D%5B%5BAll,%20-1%5D%5D;%20Rest@%20Reap%5BDo%5BMap%5BIf%5Bp%5B%23%5D%20==%200,%20Set%5Bp%5B%23%5D,%20n%5D;%20Sow%5B%23%5D%5D%20&,%20FactorInteger%5Bs%5B%5Bn%5D%5D%20%5D%5B%5B;;%20,%201%5D%5D%5D,%20%7Bn,%20Length%5Bs%5D%7D%5D%20%5D%5B%5B-1,%201%5D%5D%20%5D%20(*%20_Michael%20De%20Vlieger">] := 0; s = Import["https://oeis.org/A386482/b386482.txt", "Data"][[All, -1]]; Rest@ Reap[Do[Map[If[p[#] == 0, Set[p[#], n]; Sow[#]] &, FactorInteger[s[[n]] ][[;; , 1]]], {n, Length[s]}] ][[-1, 1]] ] (* _Michael De Vlieger, Sep 03 2025 *)

A387524 Primes p with property that in A386482 they are immediately preceded by 2p and immediately followed by 3p.

Original entry on oeis.org

3, 7, 11, 19, 31, 47, 131, 317, 947, 4327, 48091, 77237, 158489, 237733, 1973087, 3398221, 4409519

Offset: 1

N. J. A. Sloane, Sep 03 2025

This is in marked contrast to the EKG sequence A064413, where all primes > 2 have this property.

Cf. A064413, A386482, A386483, A387072, A387076, A387077, A387522, A387523.

More terms from Michael De Vlieger, Sep 03 2025

A387634 Indices k such that s(k) is a prime p, s(k-1) = 2p, and s(k+1) = 3p, where s = A386482.

Original entry on oeis.org

5, 11, 24, 31, 57, 76, 236, 479, 1532, 7652, 82123, 129791, 274542, 419569, 3367328, 7002435, 7690249

Offset: 1

Michael De Vlieger, Sep 04 2025

			Table of n, a(n), with a(n) = k, listing s(k) and its immediate neighbors:
 n  a(n) = k     s(k-1)      s(k)     s(k+1)
--------------------------------------------
 1         5         6         3          9
 2        11        14         7         21
 3        24        22        11         33
 4        31        38        19         57
 5        57        62        31         93
 6        76        94        47        141
 7       236       262       131        393
 8       479       634       317        951
 9      1532      1894       947       2841
10      7652      8654      4327      12981
11     82123     96182     48091     144273
12    129791    154474     77237     231711
13    274542    316978    158489     475467
14    419569    475466    237733     713199
15   3367328   3946174   1973087    5919261
16   7002435   6796442   3398221   10194663
17   7690249   8819038   4409519   13228557

Cf. A064413, A386482, A386483, A387072, A387076, A387077, A387522, A387523, A387524.

s = Import["https://oeis.org/A386482/b386482.txt", "Data"][[All, -1]]; m = Length[s]; t = TakeWhile[Import["https://oeis.org/A387077/b387077.txt", "Data"][[All, -1]], # <= m &]; Select[t, #1 == 2 #2 && #3 == 3 #2 & @@ s[[# - 1 ;; # + 1]] &] ]

cp's OEIS Frontend

A386482 a(1)=1, a(2)=2; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

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A386484 a(n) is the index where A387090(n) appears in A386482.

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A387522 Index of first term in A386482 that is divisible by the n-th prime.

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A387523 Primes in the order in which they first divide a term of A386482.

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A387524 Primes p with property that in A386482 they are immediately preceded by 2*p and immediately followed by 3*p.

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A387634 Indices k such that s(k) is a prime p, s(k-1) = 2*p, and s(k+1) = 3*p, where s = A386482.

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Mathematica

A387524 Primes p with property that in A386482 they are immediately preceded by 2p and immediately followed by 3p.

A387634 Indices k such that s(k) is a prime p, s(k-1) = 2p, and s(k+1) = 3p, where s = A386482.