A336957 -id:A336957 - cp's OEIS frontend

A361328 a(n) is the least prime factor of A361321(n) with a(1) = 1.

1, 2, 2, 5, 3, 3, 2, 2, 7, 3, 3, 5, 7, 2, 2, 5, 5, 2, 2, 3, 3, 7, 2, 2, 3, 7, 7, 5, 2, 2, 13, 11, 2, 2, 5, 3, 2, 2, 5, 5, 2, 2, 5, 3, 3, 7, 2, 2, 3, 3, 7, 2, 2, 3, 3, 7, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 11, 2, 2, 11, 11, 2, 2, 13, 13, 2, 2, 3, 3, 5, 7, 2, 2, 3, 5, 5

Offset: 1

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

nonn

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

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

Cf. A361321, A000469, A361327 (greatest prime factor), A361329, A361329, A336957, A360519, A361119.

A361329 a(n) = gcd(A361321(n), A361321(n+1)).

Original entry on oeis.org

1, 2, 5, 7, 3, 11, 2, 7, 13, 3, 5, 11, 7, 2, 13, 5, 17, 2, 19, 3, 17, 7, 10, 3, 23, 7, 19, 5, 2, 23, 13, 11, 2, 29, 5, 3, 2, 31, 5, 23, 2, 37, 5, 3, 29, 7, 2, 41, 3, 31, 7, 2, 43, 3, 37, 7, 2, 47, 3, 53, 2, 59, 3, 61, 2, 65, 3, 67, 2, 71, 3, 73, 2, 79, 3, 11, 17, 2, 19, 11, 23, 2, 17, 13, 19, 2

Offset: 1

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

nonn

			A361321(3) = 10, A361321(4) = 35, so a(3) = gcd(10,35) = 5;

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

Cf. A361321, A000469, A361327, A361328, A361329, A336957, A360519, A361118.

A361606 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n > 3, a(n) shares a factor with a(n-1) and a(n-2) but not with a(n-1) + a(n-2).

Original entry on oeis.org

1, 6, 10, 15, 12, 20, 45, 18, 40, 75, 24, 50, 105, 14, 30, 21, 28, 36, 63, 56, 48, 147, 98, 54, 189, 70, 60, 231, 22, 42, 33, 44, 72, 99, 88, 78, 143, 66, 26, 39, 84, 52, 91, 112, 104, 455, 80, 126, 35, 90, 154, 55, 100, 132, 135, 110, 96, 165, 130, 102, 85, 120, 34, 51, 108, 68, 153, 114

Offset: 1

Scott R. Shannon, Mar 17 2023

nonn

All terms must contain two or more distinct prime factors. If a(n) was a prime power then a(n+1) would contain the same prime factor, which in turn would imply that a(n) + a(n+1) is a multiple of the prime. But that would make finding a(n+2) impossible as any factor of a(n) would also be a factor of the sum.
To ensure the sequence is infinite a(n) must also contain a prime factor not in a(n-1). If this were not the case the sum a(n-1) + a(n) would be a multiple of the distinct prime factors of a(n), implying a(n+1) would not exist as any factor of a(n) would be a factor of the sum.
The last even term is a(114) = 210. As a(115) = 119 and a(116) = 255, the first occurrence of consecutive odd values, the resulting sum is even, so a(117) must be odd. This forces all subsequent terms to also be odd.
There is a concentration of terms at a(n) ~ 3.4*n. See the linked image. The only fixed point in the first 50000 terms is 14, although it is possible more exist.

			a(8) = 18 = 2*3*3 as a(6) = 20 = 2*2*5 and a(7) = 45 = 3*3*5 and a(6) + a(7) = 20 + 45 = 65 = 5*13. As the sum contains 5 as a factor a(8) cannot, but it must contain both 2 and 3 while containing a factor not in 45 = 3*3*5. The smallest unused number satisfying these conditions is 18.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing squarefree composites in blue, numbers neither squarefree nor prime powers in red and gold, with gold representing numbers whose prime factors have multiplicity exceeding 1.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, accentuating even terms in red.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, with a color function showing least prime factor of a(n), where red = 2, yellow = 3, green = 5, blue = 7, violet = 11, and p | a(n) such that prime p > 11 uncolored, thus, black and tiny.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, with a color function showing the number of distinct prime factors of a(n), where red = 2, yellow = 3, green = 4, and blue = 5.
Scott R. Shannon, Image of the first 50000 terms. The green line is a(n) = n.

Cf. A349493, A361102, A336957, A360519, A352867, A352774, A351001, A251622, A360209, A251604.

nn = 120; u = s = 3; c[] = False; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 6, 10}]; Set[{i, j}, {a[s - 1], a[s]}]; While[Or[c[u], PrimePowerQ[u]], u++]; Do[k = u; While[Or[c[k], CoprimeQ[i, k], CoprimeQ[j, k], ! CoprimeQ[i + j, k]], k++]; Set[{a[n], c[k], i, j}, {k, True, j, k}]; If[a[n] == u, While[Or[c[u], PrimePowerQ[u]], u++]], {n, s + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Mar 17 2023 *)

A367741 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-2) but not with a(n-1) or n.

Original entry on oeis.org

1, 3, 2, 15, 4, 5, 6, 35, 8, 7, 10, 77, 12, 11, 14, 33, 16, 55, 18, 143, 20, 13, 22, 65, 24, 25, 26, 85, 28, 17, 21, 187, 56, 99, 32, 121, 30, 253, 34, 23, 36, 115, 38, 45, 19, 39, 76, 91, 40, 49, 44, 63, 46, 119, 48, 221, 50, 51, 52, 289, 42, 323, 58, 57, 29, 95, 87, 133, 116, 171, 62, 209, 31

Offset: 1

Scott R. Shannon, Nov 29 2023

nonn

This is a variation of the Yellowstone permutation A098550 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 n+2. See the examples below.
As no term a(3*k), k>=1, can contain 3 as a factor, no term a(3*k+2) can be a power of 3 as it must share a factor with a(3*k). Likewise as a(3*(k+1)) must share a factor with a(3*k+1), the later cannot be a power of 3. Therefore no term, other than a(1), can be a power of 3, although it is likely all other positive numbers appear in the sequence.
For the terms studied, other than the first three terms and a(40) = 23 and a(45) = 19, the primes appear in their natural order.

			a(4) = 15 as a(2) = 3 which 15 shares a factor with, a(3) = 2 which 15 does not share a factor with, and 15 does not share a factor with n = 4. Also 15 has a prime factor (5) which is not a factor of 4+2 = 6. The later restriction eliminates 9 as a candidate for a(4).

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. A368231, A098550, A336957, A064413, A027748.

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.

Original entry on oeis.org

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.

A333826 a(1)=1; for n>1, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

1, 3, 6, 2, 7, 13, 20, 4, 13, 23, 34, 46, 59, 73, 88, 8, 25, 43, 62, 10, 31, 53, 76, 100, 125, 151, 178, 206, 235, 15, 46, 78, 111, 145, 5, 41, 78, 116, 155, 195, 236, 278, 321, 365, 410, 456, 503, 551, 600, 50, 101, 153, 206, 260, 315, 371, 428, 486, 545, 605, 666, 728, 791, 855, 920, 986, 1053

Offset: 1

Scott R. Shannon, Sep 03 2020

nonn

This is a variation of A337490; here we start with an offset of 1, so a(1) = 1. See that sequence for further details.

In the first 4212 terms the sequence decreases 69 times while 45 terms are repeated, the first being 13 at n=9 and the last 399876 at n=4212. After n(4166)=84 the sequence does not decrease again for n up to at least 100 million. The lowest numbers that have not appeared in that range are 30,37,47,48,49,51. The 100 millionth term is 4999999941527298.

			a(2) = 3 as the sum of all previous terms is a(1) = 1, and the GCD of 1 and 2 is 1. However 1 has already appeared so a(2) = a(1) + n = 1 + 2 = 3.
a(4) = 2 as the sum of all previous terms is a(1)+a(2)+a(3) = 10, and the GCD of 10 and 4 is 2, and as 2 has not previous appeared a(4) = 2.
a(8) = 4 as the sum of all previous terms is a(1)+...+a(7) = 52, and the GCD of 52 and 8 is 4, and as 4 has not previous appeared a(8) = 4.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Graph of the terms for n=1..4500. This includes the last known decrease in the sequence, n(4166)=84.

Cf. A337490 (same sequence rules but starting a(0)=1), A333980, A165430, A064814, A082299, A005132, A336957.

lista(nn) = {my(va = vector(nn), s=0); va[1] = 1; s += va[1]; for (n=2, nn, my(g = gcd(n, s)); if (#select(x->(x==g), va), va[n] = va[n-1]+n, va[n] = g); s += va[n];); va;} \\ Michel Marcus, Sep 05 2020

A336799 The numbers visited on a square spiral, with a(1)=1, a(2)=2, a(3) = 6, when stepping to an unvisited number as close as possible to the n = 1 starting position that has at least one common factor with the last visited number but none with the second last visited number. In case of a tie, choose the smallest number.

Original entry on oeis.org

1, 2, 6, 15, 35, 14, 12, 33, 77, 28, 10, 45, 21, 56, 20, 55, 99, 18, 34, 85, 75, 24, 22, 143, 39, 30, 46, 161, 63, 36, 40, 95, 57, 42, 26, 65, 105, 48, 38, 247, 117, 69, 115, 70, 44, 187, 51, 54, 52, 91, 119, 68, 60, 87, 203, 98, 62, 93, 129, 86, 76, 133, 175, 50, 78, 141, 235, 80, 58, 261

Offset: 1

Scott R. Shannon, Jan 27 2021

nonn

This sequence is the square spiral version of the Enots Wolley sequence A336957. The same rules for selecting the next number apply except that, instead of choosing the smallest unvisited number for a(n), the number closest to the starting n = 1 position which satisfies the selection rules is chosen. If two or more such numbers exist then the smallest is chosen.

The first term that differs from A336957 is a(9) = 77. See the examples below.

			The square spiral used is:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(8) = 33 as a(7) = 12 = 2*2*3 and a(6) = 14 = 2*7, thus a(8) must contain 3 or 2 as a factor but not 2 or 7. As a(6) excludes 2 it must contain 3 as a factor, and another prime other than 3. The closest unvisited number to the starting 1 position that satisfies these conditions is 33.
a(9) = 77 as a(8) = 33 = 3*11 and a(7) = 12 = 2*2*3, thus a(9) must contain 3 or 11 as a factor but not 2 or 3. As a(7) excludes 3 it must contain 11 as a factor, and another prime other than 11. The smallest unvisited number satisfying these conditions is 55, which is sqrt(20) ~ 4.47 units from 1. However 77 is unvisited and also satisfies the conditions, and is only 4 units from 1, thus a(9) = 77. This is the first term that differs from A336957.

Scott R. Shannon, Image of the first 100000 visited numbers on the square spiral. The colors are graduated across the spectrum from red to violet to indicate the relative visit order of the numbers. The starting 1 position is colored white. As the sequence cannot visit primes or prime powers many of the black unvisited squares form the typical diagonal-line pattern seen in the Ulam prime spiral.

Cf. A336957, A098550, A335661, A330979, A340783, A064413.

A337181 a(1) = 1, a(2) = 2; for n>2, a(n) is the smallest number not already used that is a multiple of at least one prime factor of both a(n-1) and a(n-2).

Original entry on oeis.org

1, 2, 4, 8, 12, 6, 9, 18, 24, 16, 20, 10, 25, 30, 15, 27, 36, 42, 14, 21, 28, 48, 32, 40, 44, 22, 52, 26, 56, 60, 35, 45, 50, 54, 64, 66, 68, 34, 72, 51, 63, 81, 84, 78, 39, 90, 65, 70, 75, 80, 96, 76, 38, 88, 92, 46, 100, 104, 108, 102, 99, 33, 117, 126, 91, 49, 98, 112, 116, 58, 120

Offset: 1

Scott R. Shannon, Jan 28 2021

nonn

As each term must have at least two prime factors no term, other than the initial 2, can be prime.

			a(4) = 8 as the factors of a(4-2) = a(2) = 2 and a(4-1) = a(3) = 4 = 2*2, thus a(4) must be the minimum unused multiple of 2*2 = 4, which is 8.
a(6) = 6 as the factors of a(6-2) = a(4) = 8 = 2*2*2 and a(6-1) = a(5) = 12 = 2*2*3, thus a(4) must be the minimum unused multiple of 2*2 = 4 or 2*3 = 6. As 4 has been used a(6) = 6.
a(13) = 25 as the factors of a(13-2) = a(11) = 20 = 2*2*5 and a(13-1) = a(12) = 10 = 2*5, thus a(13) must be the minimum unused multiple of 2*2 = 4, 2*5 = 10, or 5*5 = 25. As 4,8,10,12,16,20,24 have been used, a(13) = 25.

Scott R. Shannon, Image of the terms for n=1..500000. The green line is a(n) = n, indicating the other lines are slightly curved downward.

Cf. A064413, A336957, A098550, A270139.

A338055 Lexicographically earliest infinite sequence {a(n)} of distinct positive numbers such that, for n>2, a(n) has a common factor with a(n-1) involving only primes <= 11 but no such common factor with a(n-2) (primes > 11 play no role in this definition).

Original entry on oeis.org

1, 2, 6, 15, 35, 14, 12, 33, 55, 10, 18, 21, 77, 22, 20, 45, 63, 28, 40, 75, 99, 44, 50, 105, 231, 88, 80, 135, 147, 56, 100, 165, 189, 98, 110, 225, 441, 112, 160, 275, 297, 24, 70, 385, 363, 36, 140, 539, 891, 30, 175, 847, 66, 60, 245, 3773, 132, 90, 875, 5929, 176, 48

Offset: 1

N. J. A. Sloane, Oct 11 2020, based on an email from Frank Stevenson, Aug 26 2020

nonn

Let p_i denote the i-th prime. If the prime decompositions of x and y are
   x = Product_{i=1..5} p_i^e_i*q_x, y = Product_{i=1..5} p_i^f_i*q_y,
then we define gcd_11(x, y) to be Product_{i=1..5} p_i^min{e_i, f_i}.
The sequence is the lexicographically earliest infinite sequence {a(n)} of distinct positive numbers such that, for n>2, gcd_11(a(n), a(n-1)) > 1 and gcd_11(a(n), a(n-2)) = 1.
An analog of A336957, but using only the first five primes.
Frank Stevenson has proved that a(n) always exists, something that is not true if only the primes 2, 3, 5, 7 are used.  He remarks that because the small primes 13, 17, 19, ... cannot be used in the construction, some numbers take a long time to appear - are very late, in the terminology of A338053.
As can be seen from the graph, this is a much more irregular sequence than A336957.

Frank Stevenson, Table of n, a(n) for n = 1..843

Cf. A336957, A338053, A338054.

A338058 A337645(n)/2.

Original entry on oeis.org

1, 3, 5, 5, 5, 5, 5, 5, 5, 9, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 18, 18, 18, 18, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 27, 27, 27, 27, 33, 33, 33, 33, 36

Offset: 1

N. J. A. Sloane, Oct 17 2020

nonn

The smallest missing legal number in A336957, divided by 2.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A336957, A337645.

cp's OEIS Frontend

A361328 a(n) is the least prime factor of A361321(n) with a(1) = 1.

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A361329 a(n) = gcd(A361321(n), A361321(n+1)).

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A361606 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n > 3, a(n) shares a factor with a(n-1) and a(n-2) but not with a(n-1) + a(n-2).

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Mathematica

A367741 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-2) but not with a(n-1) or n.

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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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A333826 a(1)=1; for n>1, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

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A337181 a(1) = 1, a(2) = 2; for n>2, a(n) is the smallest number not already used that is a multiple of at least one prime factor of both a(n-1) and a(n-2).

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A338055 Lexicographically earliest infinite sequence {a(n)} of distinct positive numbers such that, for n>2, a(n) has a common factor with a(n-1) involving only primes <= 11 but no such common factor with a(n-2) (primes > 11 play no role in this definition).

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A338058 A337645(n)/2.

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