A347179 -id:A347179 - cp's OEIS frontend

A347406 Earliest sequence of distinct positive integers such that both gcd(a(n),a(n-k)) = 1 and gcd(a(n),a(n+k)) = 1, where k is each divisor of a(n) and n - k >= 1.

1, 2, 3, 5, 4, 7, 9, 8, 11, 13, 10, 17, 19, 14, 15, 23, 16, 29, 21, 26, 27, 25, 22, 31, 35, 32, 33, 37, 38, 41, 39, 34, 43, 47, 28, 53, 51, 20, 57, 59, 40, 61, 49, 44, 63, 67, 46, 71, 73, 52, 69, 79, 50, 83, 81, 55, 58, 77, 65, 82, 87, 85, 89, 74, 93, 95, 91, 86, 97, 101, 62, 103, 45, 64, 75

Offset: 1

Scott R. Shannon, Aug 30 2021

The majority of terms are concentrated along two lines, the upper line has gradient of approximately 1.37 while the lower line has a gradient of approximately 1.02. Between these a third more random line also appear. See the linked image.

Small numbers with only 2 and 3 as prime divisors apparently take many terms to appear. For example a(210613) = 6, a(224221) = 18, while 12 and 24 have not appeared after 250000 terms.

			a(3) = 3 as the divisors of 3 are 1 and 3, and a(3-1) = a(2) = 2, a(3+1) = a(4) = 5, and a(3+3) = a(6) = 7, and the gcd of 3 and each of these three numbers is 1. As a(3-3) = a(0) is not defined this term is ignored.
a(11) = 10 as the divisors of 10 are 1, 2, 5 and 10, and a(11-1) = a(10) = 13, a(11-2) = a(9) = 11, a(11-5) = a(6) = 7, a(11-10) = a(1) = 1, a(11+1) = a(12) = 17, a(11+2) = a(13) = 19, a(11+5) = a(16) = 23, and a(11+10) = a(21) = 27, and the gcd of 10 and each of these eight numbers is 1.
a(13) = 19 as the divisors of 19 are 1 and 19, and a(13-1) = a(12) = 17, a(13+1) = a(14) = 14, and a(13+19) = a(32) = 34, and the gcd of 19 and each of these three numbers is 1. Note that as a(11) = 10, and a(11+2) = a(13), where 2 is a divisor of 10, a(13) cannot equal 15 as gcd(10,15) > 1. This is the first term that differs from A347179.

Scott R. Shannon, Image of the first 250000 terms.

Cf. A347179, A027750, A000005, A084937, A098550, A336957.

A347404 a(1) = 3; for n > 2, a(n) is the smallest distinct positive integer such that gcd(a(n), a(n-k)) > 1 for each divisor k of a(n) such that k < n.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 10, 2, 14, 4, 22, 8, 24, 16, 20, 28, 26, 30, 34, 32, 36, 40, 5, 50, 60, 38, 42, 44, 46, 48, 54, 52, 58, 56, 62, 64, 66, 68, 70, 72, 74, 78, 39, 90, 57, 21, 51, 69, 84, 87, 33, 27, 96, 81, 63, 7, 126, 45, 168, 99, 93, 75, 95, 19, 114, 105, 77, 198, 153, 165, 11, 132, 55, 595

Offset: 1

Scott R. Shannon, Aug 30 2021

nonn

The majority of terms are concentrated along a line whose slope is approximately 1.3. Occasionally though there are terms which correspond to the smallest unused number up to that point, and these tend to lead to a subsequent very large term. For example a(499) = 628, a(500) = 682, a(501) = 31, a(502) = 14322. Other large terms appear seemingly at random, for example a(15449) = 19880, a(15450) = 19099, a(15451) = 74962230.
It is likely all numbers > 1 eventually appear. The smallest number not seen after 20000 terms is 89.
Note that if the sequence starts with 2 then the terms are just all the increasing even numbers.

			a(2) = 6. As a(1) = 3 the next term must be a multiple of 3, and the smallest unused such number is 6. Note that as a(2-2), a(2-3) and a(2-6) are not defined these are ignored.
a(7) = 10. As a(6) = 18 the next term must have 2 and/or 3 as divisors. If it has 2 as a divisor is must also have 3 and/or 5 as a divisor as a(8-2) = a(6) = 15. The smallest unused number satisfying these is 10. Note that as 5 is a divisor of 10 it must be that a(7-5) = a(2) = 6 has 2 or 5 as a divisor, which is true.
a(8) = 2. As a(7) = 10 the next term must have 2 and/or 5 as a divisor. As a(6) = 18 also has 2 as a divisor a(8) = 2 is the next smallest unused term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..15450, showing records in red, local minima in blue, primes in green, highlighting fixed points in amber.

Cf. A347179, A347619, A347406, A027750, A098550, A336957.

nn = 74; c[] = 0; a[1] = 3; c[3] = 1; u = 2; While[c[u] > 0, u++]; Do[k = u; While[Nand[c[k] == 0, AllTrue[TakeWhile[Divisors[k], # < i &], ! CoprimeQ[a[i - #], k] &]], k++]; Set[{a[i], c[k]}, {k, i}], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 12 2022 *)

cp's OEIS Frontend

A347406 Earliest sequence of distinct positive integers such that both gcd(a(n),a(n-k)) = 1 and gcd(a(n),a(n+k)) = 1, where k is each divisor of a(n) and n - k >= 1.

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