A367741 -id:A367741 - cp's OEIS frontend

A368231 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-1) but not with a(n-2) or n.

1, 15, 35, 77, 143, 65, 30, 21, 91, 221, 85, 55, 33, 39, 182, 133, 95, 115, 69, 51, 170, 145, 203, 119, 102, 45, 155, 341, 154, 161, 207, 57, 190, 185, 407, 187, 153, 63, 217, 403, 130, 205, 123, 87, 319, 209, 247, 299, 138, 93, 589, 323, 238, 259, 111, 75, 70, 287, 451, 253, 230, 195, 377

Offset: 1

Scott R. Shannon, Dec 18 2023

nonn

This is a variation of the Enots Wolley sequence A336957 and A360519, with an additional restriction that no term a(n) can have a common factor with n. For the sequence to be infinite a(n) must always have a prime factor that is not a factor of a(n-1)*(n+1). See the examples below.

Other than no term being a prime or prime power, see A336957, no term can be an even number with only two distinct prime factors. Clearly no term a(2*k) can be even, so if we assume that a(2*k+1) = 2^n*p^m, with n and m>=1, then a(2*k) must have p as a factor. But as a(2*k+2) must share a factor with a(2*k+1) and cannot have 2 as a factor, it must also have p as a factor. However that is not allowed as a(n) cannot share a factor with a(n-2), so no term can be even with only two distinct prime factors. Therefore the smallest even number is a(7) = 30.

			a(2) = 15 as 15 is the smallest number that is not a prime power and does not have 2 as a factor.
a(3) = 35 as a(3) is chosen so it shares a factor with a(2) = 3*5 while not having 3 as a factor; it therefore must be a multiple of 5 while not being a power of 5. The smallest number meeting those criteria is 10, but a(2)*(3+1) = 15*4 = 60, and 10 has no prime factor not in 60, so choosing 10 would mean a(4) would not exist. The next smallest available number is 35.
a(4) = 77 as a(4) must be a multiple of 7 but not a power of 7, not a multiple of 2, 3 or 5, while having a prime factor not in 35*(4+1) = 165. The smallest number satisfying these criteria is 77.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A367741, A336957, A360519, A098550, A064413, A027748.

A367742 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-2) and n but not with a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 15, 8, 21, 10, 9, 5, 33, 20, 39, 14, 27, 16, 51, 22, 57, 26, 45, 28, 69, 32, 75, 34, 63, 38, 87, 40, 93, 44, 81, 46, 105, 52, 111, 50, 99, 25, 123, 35, 129, 55, 6, 115, 94, 135, 56, 65, 12, 13, 106, 117, 80, 91, 18, 203, 118, 145, 122, 155, 183, 62, 165, 58, 201, 64, 141, 68, 213

Offset: 1

Scott R. Shannon, Nov 29 2023

nonn

This is a variation of the Yellowstone permutation A098550 with an additional restriction that each term a(n) must have a common factor with n. For the sequence to be infinite a(n) must be chosen so it does not have as factors all the prime factors of n+1. See the examples below.

Unlike A098550 the primes do not appear in their natural order, and in general are delayed in their appearance relative to similarly sized numbers. In the first 100000 terms the fixed points are 1, 2, 3, 4, 9, 14, 16, 74, 76, 86, 88, 207, 320, 322, 901; it is unknown if more exist. The sequence is conjectured to be a permutation of the positive numbers.

			a(5) = 15 as 15 shares a factor with a(3) = 3 and with n = 5, does not share a factor with a(4) = 4, and 15 does not have as factors all the prime factors of 5+1 = 6 = 2*3.
a(55) = 80 as 80 shares a factor with a(53) = 106 and with n = 55, does not share a factor with a(54) = 117, and 80 does not have as factors all the prime factors of 55+1 = 56 = 2^3*7. Note that 70 satisfies the first three criteria but not the last, so choosing a(55) = 70 would mean a(56) would not exist.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms for a(n) <= 500000. The green line is a(n) = n.

Cf. A367741, A098550, A336957, A064413, A027748, A368231.

cp's OEIS Frontend

A368231 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-1) but not with a(n-2) or n.

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