A336957 -id:A336957 - cp's OEIS frontend

A337182 a(1) = 1, a(2) = 2; for n>2, a(n) is the smallest number not already used which is a multiple of the product of the most frequently occurring distinct prime factors in a(n-1).

1, 2, 4, 6, 12, 8, 10, 20, 14, 28, 16, 18, 3, 9, 15, 30, 60, 22, 44, 24, 26, 52, 32, 34, 68, 36, 42, 84, 38, 76, 40, 46, 92, 48, 50, 5, 25, 35, 70, 140, 54, 21, 63, 27, 33, 66, 132, 56, 58, 116, 62, 124, 64, 72, 74, 148, 78, 156, 80, 82, 164, 86, 172, 88, 90, 39, 117, 45, 51, 102, 204

Offset: 1

Scott R. Shannon, Jan 29 2021

nonn

The terms are concentrated along lines, similar to A098550 and A336957, but show more complicated behavior. The sparse lines show wavy and random variations, some approaching each other before separating again. More intriguingly the data up to 3.5 million terms show at least three well defined lines that curve slowly upward and cross other lines which they initially start below. See the first and second linked image. Interestingly, terms with 3 and 7 as factors do not appear to be part of these three curving lines, while other primes examined up to 19 are part of these lines. See the last link images.
The even terms dominate the higher values for a given range of n. The image for the next least prime factor of these even terms shows terms with 2 and 3 spread across multiple lines, some well defined and others sparse. Terms with 2 and 5 are in only four lines along with two other lines that appear to contain terms with 2 and all other primes as factors. See the third linked image.
The lower terms for a given range of n are predominantly those with only one, two or three factors, while as the values increase they tend to have more and more factors. However, there are several lines on which terms with different numbers of factors all lie along. See the fourth linked image.
For the first 3.5 million terms the primes appear in their natural order. See A338222 for the index where the primes appear. Although unproven it is highly probable that this is true for all primes. It is also likely all numbers eventually appear, i.e., this is a permutation of the positive integers. The lowest unseen number after 3.5 million terms is 1523. In the same range the only fixed point, other than the initial terms 1 and 2, is 15.
There are at least four lines on the graph of the first 20000 terms. They can be characterized as follows, from the highest sloped L1 to the lowest sloped L4, considering terms within 1% of the fitted equations. The approximate slopes of the 4 lines are 2.65733, 1.9927, 1.3286 and 0.66399, so that the normalized slopes of L1 thru L4 are 4, 3, 2 and 1 respectively. L1 and L3 terms consist of even values, L4 terms consist of odd values. The four lines encompass approx. 75% of the 20000 terms. - Bill McEachen, Aug 15 2025

			a(4) = 6 as a(3) = 4 = 2*2, and since 2 is the only prime factor, a(4) must be the smallest unused multiple of 2, which is 6.
a(5) = 12 as a(4) = 6 = 2*3, thus as 2 and 3 both occur once, a(5) must be the smallest unused multiple of 2*3 = 6, which is 12.
a(6) = 8 as a(5) = 12 = 2*2*3, thus as 2 is the most frequently occurring factor, a(6) must be the smallest unused multiple of 2, which is 8.
a(13) = 3 as a(12) = 18 = 2*3*3, thus as 3 is the most frequently occurring factor, a(13) must be the smallest unused multiple of 3, which is 3.
a(17) = 60 as a(16) = 30 = 2*3*5, thus as 2,3 and 5 all occur once, a(17) must be the smallest unused multiple of 2*3*5 = 30, which is 60.

Scott R. Shannon, Table of n, a(n) for n = 1..20000
Scott R. Shannon, Image of the first 3500000 terms. The green line is a(n)=n.
Scott R. Shannon, Image of the first 3500000 terms showing the least prime factor. The terms with at least prime factor of 2,3,5,7,11,13 or 17 are shown as red, orange, yellow, green, blue, indigo and violet respectively, while terms with a least prime factor >= 19 are shown as white.
Scott R. Shannon, Image of the first 3500000 terms showing the least prime factor other than 2 for the even terms. The even terms with a next least prime factor of 3,5,7,11,13 or 17 are shown as orange, yellow, green, blue, indigo and violet respectively, while terms with a least prime factor >= 19 are shown as white.
Scott R. Shannon, Image of the first 3500000 terms showing the number of prime divisors for each term. The terms with 1,2,3,4,5,6,7 or 8 prime factors are shown as white, red, orange, yellow, green, blue, indigo and violet respectively, while terms with 9 or more are show in grey.
Scott R. Shannon, Image of the first 3500000 terms with 3 as a factor. Note none of the lines are curved.
Scott R. Shannon, Image of the first 3500000 terms with 5 as a factor.
Scott R. Shannon, Image of the first 3500000 terms with 7 as a factor. Note none of the lines are curved.
Scott R. Shannon, Image of the first 3500000 terms with 11 as a factor.
Scott R. Shannon, Image of the first 3500000 terms with 13 as a factor.
Scott R. Shannon, Image of the first 3500000 terms with 17 as a factor.
Scott R. Shannon, Image of the first 3500000 terms with 19 as a factor.

Cf. A338222 (index of the prime terms), A098550, A336957, A064413, A027746.

A338052 a(n) = A337645(n-1) - n.

Original entry on oeis.org

0, 3, 6, 5, 4, 3, 2, 1, 0, 7, 8, 7, 6, 5, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, 0, -1, -2, -3, 4, 3, 2, 1, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -14, -15, -16, -17, -6, -7, -8, -9, -4, -5, -6, -7, -2, -3, -4, -5, 2, 1, 0

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Oct 11 2020

sign

Motivation: This is (smallest missing legal number in A336967) - n. For when we are trying to find A336957(n), the smallest legal possibility is A337645(n-1), which appears to grow like n.

However, the conclusion from looking at the graph of the present sequence is that A337645(n) ~ n*(1 - 1/50) or perhaps n*(1 - c/log n) would be a better approximation to A337645.

N. J. A. Sloane, Table of n, a(n) for n = 2..20001

A338060 a(n) = m such that A338059(m) = n, or -1 if n does not appear in A338059.

Original entry on oeis.org

1, 2, 5, 3, 8, 4, 11, 13, 6, 19, 17, 14, 29, 12, 7, 20, 34, 21, 38, 26, 22, 25, 41, 51, 9, 30, 15, 31, 58, 47, 64, 36, 16, 35, 10, 55, 73, 37, 28, 43, 79, 46, 92, 61, 27, 42, 97, 90, 23, 60, 33, 66, 133, 95, 18, 50, 39, 70, 136, 85, 139, 65, 32, 75, 44, 100, 144, 76, 40, 69

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 18 2020

nonn

It is conjectured that A338059 is a permutation of the positive numbers. This is (conjecturally) the inverse permutation.

Cf. A336957, A338059.

A347179 a(1) = 1; for n > 1, a(n) = smallest distinct positive integer such that gcd(a(n),a(n-k)) = 1, where k is each divisor of a(n) and n - k >= 1.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 8, 11, 13, 10, 17, 15, 14, 19, 23, 16, 21, 25, 26, 29, 31, 22, 27, 37, 20, 41, 33, 28, 39, 43, 32, 47, 49, 34, 45, 53, 35, 58, 51, 59, 46, 61, 55, 57, 62, 65, 67, 69, 38, 71, 73, 50, 77, 79, 64, 75, 83, 44, 85, 81, 76, 87, 89, 56, 97, 63, 68, 91, 95, 74, 93, 101, 52, 103

Offset: 1

Scott R. Shannon, Aug 21 2021

The majority of terms are concentrated along two lines, the upper line has gradient of approximately 1.342, while the lower line, which is less well defined, has a gradient of approximately 1.05. See the linked image.

Small numbers with only 2 and 3 as prime divisors apparently take many terms to appear. For example a(64963) = 6, a(80415) = 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) = 2 which has no common divisor with 3. As a(3-3) = a(0) is not defined this term is ignored.
a(5) = 4 as the divisors of 4 are 1, 2 and 4, and a(5-1) = a(4) = 5, a(5-2) = a(3) = 3, and a(5-4) = a(1) = 1, and the gcd of 4 and these three numbers is 1.
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, and a(11-10) = a(1) = 1, and the gcd of 10 and these four numbers is 1.

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

Cf. A347406, A027750, A000005, A084937, A098550, A336957.

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.

Original entry on oeis.org

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.

A350806 The maximum value reached when starting at n for the iteration map A350805(n).

Original entry on oeis.org

1, 2, 1385, 4, 16, 160, 1385, 1385, 32, 1385, 160, 12, 268, 1385, 124, 16, 1385, 32, 268, 32, 32, 1385, 1385, 268, 32, 124, 30, 268, 124, 30, 1385, 32, 1385, 160, 1385, 1385, 124, 1385, 1385, 268, 124, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 1385, 160, 1385, 1385, 56, 1385

Offset: 1

Scott R. Shannon, Jan 17 2022

nonn

For smaller values of n a maximum value of 1385 is common; this is similar to the value 9232 for the standard Collatz map. See A025586.

A graph of the terms looks similar to those of A336957 and A098550. See the linked image.

			a(6) = 160 as the map A350805(6) gives 6 -> 11 -> 34 -> 53 -> 160 -> 21 -> 32 -> 5 -> 16 -> 4 -> 2 -> 1, the maximum value of which is 160.

Scott R. Shannon, Image of the first 10 million terms.

Cf. A350805, A025586, A006370, A027746, A350807, A006577, A000040, A336957, A098550.

A350927 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1) * |a(n-1) - a(n-2)|, a(n)) > 1.

Original entry on oeis.org

1, 2, 4, 6, 3, 9, 8, 10, 5, 15, 12, 14, 7, 21, 16, 18, 20, 22, 11, 33, 24, 26, 13, 39, 27, 28, 30, 25, 35, 32, 34, 17, 51, 36, 38, 19, 57, 40, 42, 44, 46, 23, 69, 45, 48, 50, 52, 54, 56, 49, 63, 58, 29, 87, 60, 55, 65, 62, 31, 93, 64, 66, 68, 70, 72, 74, 37, 111, 75, 76, 78, 80, 82, 41, 123, 81

Offset: 1

Scott R. Shannon, Jan 28 2022

nonn

This is likely a permutation of the natural numbers. The sequence shows similar behavior to the EKG sequence A064413. In the first 500000 terms the fixed points are 1, 2, 77, 221, and it is likely no more exist.

			a(4) = 6 as a(3)*|a(3)-a(2)| = 4*2 = 8, 6 has not been used and gcd(6,8) > 1.
a(8) = 10 as a(7)*|a(7)-a(6)| = 8*1 = 8, 10 has not been used and gcd(10,8) > 1.

Scott R. Shannon, Image of the first 10000 terms. The green line is y = n.

Cf. A351001, A337136, A064413, A336957, A098550, A089088, A251622.

A355061 Lexicographically earliest infinite sequence of positive numbers such that, for n>2, a(n) has a common factor with a(n-1), no common factor with a(n-2), and the product a(n)a(n-1) is distinct from all previous products, a(i)a(i-1), i=2..n-1.

Original entry on oeis.org

1, 2, 6, 15, 35, 14, 6, 33, 55, 10, 6, 21, 35, 10, 12, 21, 77, 22, 6, 39, 65, 10, 14, 21, 15, 10, 22, 33, 15, 20, 14, 63, 15, 40, 14, 77, 33, 12, 14, 91, 39, 12, 20, 35, 63, 6, 26, 65, 15, 12, 22, 55, 15, 18, 28, 35, 45, 12, 26, 91, 21, 30, 22, 143, 39, 15, 50, 22, 99, 15, 70, 22, 187, 51, 6

Offset: 1

Scott R. Shannon, Jun 16 2022

nonn

Like the Enots Wolley sequence, A336957, no term a(n) can be a prime or a prime power as this would make it impossible to find a(n+1). As 6 is the smallest number to include two different primes, and hence the smallest number beyond the first two terms that can appear, it occurs frequently in the sequence, 1887 times in the first 250000 terms. See A355139 for the indices of these terms.
Unlike A336957 multiple odd successive terms occur, the longest such run in the first 250000 terms being fourteen starting at a(111799) = 20257.
See A355138 for the products of consecutive terms.

			a(5) = 35 as this is the smallest number to share a factor with a(4) = 15, not share a factor with a(3) = 6, and contains a prime factor not in a(4) = 15 and hence allows a(6) to exist.
a(7) = 6 as this is the smallest number to share a factor with a(6) = 14, not share a factor with a(5) = 35, and contains a prime factor not in a(6) = 14 and hence allows a(8) to exist. This is the first term to differ from A336957.

Scott R. Shannon, Image of the first 250000 terms. The green line is y = n.

Cf. A355138, A355139, A336957, A098550, A064413.

from sympy import primefactors
from itertools import count, islice
def agen(): # generator of terms
    an1, an, f1, f, pset = 2, 6, {2}, {2, 3}, {2, 12}
    yield from [1, 2, 6]
    for n in count(4):
        an2, an1, an, f2, f1 = an1, an, 6, f1, f
        f = set(primefactors(an))
        while an*an1 in pset or f1&f == set() or f2&f != set() or f <= f1:
            an += 1; f = set(primefactors(an))
        pset.add(an*an1); yield an
print(list(islice(agen(), 75))) # Michael S. Branicky, Jun 20 2022

A356903 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous tau(a(n)) terms.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 9, 11, 13, 17, 8, 15, 19, 23, 29, 14, 25, 27, 31, 37, 22, 35, 39, 41, 43, 34, 47, 21, 53, 55, 26, 49, 51, 59, 61, 10, 67, 33, 71, 73, 38, 65, 69, 77, 79, 58, 83, 57, 85, 89, 46, 91, 87, 95, 97, 62, 101, 103, 81, 107, 74, 109, 113, 93, 115, 82, 119, 121, 111, 125, 86, 127, 131, 123

Offset: 1

Scott R. Shannon, Sep 03 2022

nonn

The terms are concentrated along various lines that contain numbers with a lowest prime factor of 2, 3 or 5. These lines appear to have a slight upward curvature. However the uppermost line, which has a gradient of ~1.22, contains numbers with all prime factors. See the linked images.

Numbers with a large number of divisors relative to the numbers close to it appear much later in the sequence. For example a(96) = 6, a(1873) = 12, a(2328) = 18, a(192) = 16. The sequence is conjectured to be a permutation of the positive integers although it may take a very large number of terms for some values to appear, e.g., after 500000 terms numbers such as 24, 30, 36 have not occurred. In the same range the longest run of consecutive odd values is seven, while the only fixed points are the first three terms, although it is possible others exist for very large values of n if the smaller terms continue to increase relative to the uppermost line.

			a(7) = 9 as tau(9) = A000005(9) = 3, and 9 is coprime to the previous three terms, namely a(6) = 4, a(5) = 7 and a(4) = 5.

Scott R. Shannon, Image of the first 500000 terms.
Scott R. Shannon, Image of the first 500000 terms with color. The terms with a lowest prime factor of 2 are shown in red, those with 3 are shown in yellow, those with 5 are shown in green, while all others are shown in white.

Cf. A356850, A000005, A356851, A093714, A336957, A000040.

A361327 a(n) is the greatest prime factor of A361321(n) with a(1) = 1.

Original entry on oeis.org

1, 3, 5, 7, 7, 11, 11, 7, 13, 13, 5, 11, 11, 7, 13, 13, 17, 17, 19, 19, 17, 17, 7, 5, 23, 23, 19, 19, 11, 23, 23, 13, 11, 29, 29, 7, 13, 31, 31, 23, 23, 37, 37, 11, 29, 29, 11, 41, 41, 31, 31, 13, 43, 43, 37, 37, 17, 47, 47, 53, 53, 59, 59, 61, 61, 13, 13, 67, 67, 71, 71, 73, 73, 79, 79, 11, 17

Offset: 1

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 15 2023

nonn

			A361321(3) = 10 = 2*5, so a(3) = 5.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A361321, A000469, A361328 (least prime factor), A361329, A361329, A336957, A360519, A361120.

cp's OEIS Frontend

A337182 a(1) = 1, a(2) = 2; for n>2, a(n) is the smallest number not already used which is a multiple of the product of the most frequently occurring distinct prime factors in a(n-1).

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A338052 a(n) = A337645(n-1) - n.

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A338060 a(n) = m such that A338059(m) = n, or -1 if n does not appear in A338059.

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A347179 a(1) = 1; for n > 1, a(n) = smallest distinct positive integer such that gcd(a(n),a(n-k)) = 1, where k is each divisor of a(n) and n - k >= 1.

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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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A350806 The maximum value reached when starting at n for the iteration map A350805(n).

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A350927 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1) * |a(n-1) - a(n-2)|, a(n)) > 1.

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A355061 Lexicographically earliest infinite sequence of positive numbers such that, for n>2, a(n) has a common factor with a(n-1), no common factor with a(n-2), and the product a(n)*a(n-1) is distinct from all previous products, a(i)*a(i-1), i=2..n-1.

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A356903 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous tau(a(n)) terms.

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A361327 a(n) is the greatest prime factor of A361321(n) with a(1) = 1.

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A355061 Lexicographically earliest infinite sequence of positive numbers such that, for n>2, a(n) has a common factor with a(n-1), no common factor with a(n-2), and the product a(n)a(n-1) is distinct from all previous products, a(i)a(i-1), i=2..n-1.