A366111 -id:A366111 - cp's OEIS frontend

A365984 Starting with a(1) = 2, the lexicographically earliest infinite sequence of distinct positive integers such that |a(n) - a(n-1)| is a divisor of a(n), and where |a(n) - a(n-1)| > 1.

2, 4, 6, 8, 10, 12, 9, 18, 15, 20, 16, 14, 21, 24, 22, 33, 30, 25, 50, 40, 32, 28, 26, 39, 36, 27, 54, 45, 42, 35, 70, 56, 48, 44, 46, 69, 66, 55, 60, 57, 38, 76, 72, 63, 84, 77, 88, 80, 64, 62, 93, 90, 75, 78, 52, 65, 130, 104, 91, 98, 96, 92, 94, 141, 138, 115, 110, 99, 102, 51, 34, 68, 85, 170

Offset: 1

Scott R. Shannon, Sep 24 2023

nonn

For the sequence to be infinite no term can be a prime except for a(1) = 2. One can easily show that if a(n) is a prime p, then the only possible value for a(n-1) or a(n+1) is 2p. If a(n) = p was a term then the difference between it and the previous term must also be p, implying the previous term is a multiple of p, so it must be 2p. As 2p has now already appeared the term after p would not exist, thus terminating the sequence.

The first term that is not a prime power that cannot be used even though it satisfies being divisible by the difference between it and the previous term is 175, which appears to be a valid value for a(214) since a(213) = 350. However the next term after 175 would have to be one of 140, 150, 168, 170, 180, 182, 200, 210, 350, but all of those values have already appeared as previous terms, so 175 can never appear else it would terminate the sequence.

Scott R. Shannon, Table of n, a(n) for n = 1..5000
a(4) = 8 as |8 - a(3)| = |8 - 6| = 2, and 2 is a divisor of 8. Note that 3 would also satisfy this requirement, but as shown above a prime will terminate the sequence so is not permitted.

Cf. A027750, A366111, A363576, A359799.

A366047 Starting with a(1) = 2, the lexicographically earliest infinite sequence of distinct positive integers such that |a(n) - a(n-1)| is a divisor of a(n)*a(n-1), where |a(n) - a(n-1)| is not a prime and greater than 1.

Original entry on oeis.org

2, 6, 10, 14, 18, 9, 36, 12, 4, 8, 16, 20, 24, 15, 30, 21, 42, 28, 32, 40, 44, 22, 26, 52, 48, 39, 78, 60, 35, 70, 45, 54, 27, 108, 72, 56, 64, 68, 34, 38, 76, 80, 55, 110, 66, 33, 132, 84, 63, 90, 65, 130, 104, 88, 92, 46, 50, 25, 150, 75, 100, 96, 87, 174, 58, 62, 124, 116, 112, 98, 49, 392

Offset: 1

Scott R. Shannon, Sep 27 2023

nonn

For the sequence to be infinite no term can be a prime except for a(1) = 2. One can show that if a(n) is a prime p, then the only possible value for a(n-1) is 2p or p + p^2 since, if a term is prime, the preceding term must be a multiple of that prime. However the preceding term cannot be 2p since the difference between the terms would then be prime, therefore it must be p + p^2. However the only possible value for the term after a prime p is likewise p + p^2, but that has already been used, thus allowing a term to be prime would terminate the sequence.

			a(9) = 4 as |4 - a(8)| = |4 - 12| = 8, and 8 is a divisor of 4*12 = 48 and is not a prime. Note that |3 - 12| = 9 is a divisor of 3*12 = 36 and is not a prime, but as shown above a prime term will terminate the sequence so is not permitted.

Scott R. Shannon, Table of n, a(n) for n = 1..10000.
Michael De Vlieger, Log log scatterplot of a(n) n = 1..1024, showing prime powers in gold, squarefree numbers in green, and numbers neither squarefree nor prime powers in blue, highlighting powerful numbers that are not prime powers in light blue.

Cf. A027750, A366111, A365984, A363576, A359799.

nn = 120; c[_] := False; s = {2, 6};
f[x_] := Times @@ FactorInteger[x][[All, 1]];
MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, s];
Set[{j, u}, {s[[-1]], 4}];
Do[k = u;
    While[Or[c[k], #1 < 4, PrimeQ[#1],
          ! Divisible[j*k, #1], ! Divisible[j, #2], ! Divisible[k, #2]] & @@
          {#, f[#]} &@ Abs[j - k], k++];
    Set[{a[n], c[k], j}, {k, True, k}];
    If[k == u, While[Or[c[u], PrimeQ[u]], u++]], {n, Length[s] + 1, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 29 2023 *)

cp's OEIS Frontend

A365984 Starting with a(1) = 2, the lexicographically earliest infinite sequence of distinct positive integers such that |a(n) - a(n-1)| is a divisor of a(n), and where |a(n) - a(n-1)| > 1.

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