A349472 -id:A349472 - cp's OEIS frontend

A349493 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-2)+a(n-1), a(n)) > 1 while gcd(a(n-2), a(n)) = 1 and gcd(a(n-1), a(n)) = 1.

1, 2, 3, 5, 4, 9, 13, 8, 7, 15, 11, 14, 25, 27, 16, 43, 59, 6, 35, 41, 12, 53, 55, 18, 73, 49, 10, 177, 17, 20, 37, 19, 21, 22, 215, 39, 28, 67, 45, 26, 71, 97, 24, 77, 101, 30, 131, 23, 32, 33, 65, 34, 57, 91, 40, 393, 433, 38, 51, 89, 44, 63, 107, 46, 75, 121, 52, 173, 69, 50, 119, 117, 58, 85

Offset: 1

Scott R. Shannon, Nov 20 2021

nonn

In the first 100000 terms the smallest unseen number is 14657, although it is likely all numbers eventually appear. In the same range the fixed points are 3, 8, 11, 69, 207, 543, 555, 663, 687, 981. The majority of terms more than n = 100000 appear to move away from the line y = n, see the linked image, so it is unclear if more exist. The largest value in the first 100000 terms is a(87952) = 4758245.

			a(3) = 3 as a(1)+a(2) = 3, gcd(1,3) = 1, gcd(2,3) = 1, gcd(3,3) > 1 and 3 is unused.
a(4) = 5 as a(2)+a(3) = 5, gcd(2,5) = 1, gcd(3,5) = 1, gcd(5,5) > 1 and 5 is unused.
a(8) = 8 as a(6)+a(7) = 22, gcd(9,8) = 1, gcd(13,8) = 1, gcd(22,8) > 1 and 8 is unused.

Scott R. Shannon, Image of the first 100000 terms for values less than 150000. The green line is y = n.
Scott R. Shannon, Image of the first 100000 terms.

Cf. A349492, A119018, A064413, A349472.

a[1]=1; a[2]=2; a[n_]:=a[n]=(k=2;While[MemberQ[Array[a,n-1],k]||GCD[a[n-2]+a[n-1],k]<=1||GCD[a[n-2],k]!=1||GCD[a[n-1],k]!=1,k++];k); Array[a,74] (* Giorgos Kalogeropoulos, Nov 20 2021 *)

from math import gcd
terms, appears = [1, 2], {2:True}
for n in range(3, 100):
    t = 3
    while not(appears.get(t) is None and gcd(terms[-2]+terms[-1], t)>1 and gcd(terms[-2], t)==1 and gcd(terms[-1], t)==1):
        t += 1
    appears[t] = True; terms.append(t);
print(terms) #Gleb Ivanov, Nov 20 2021

A349492 a(1)=1, a(2)=6; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1)+n, a(n)) > 1, gcd(a(n-1), a(n)) > 1, and gcd(n, a(n)) > 1.

Original entry on oeis.org

1, 6, 3, 42, 470, 2, 84, 4, 78, 8, 418, 10, 598, 12, 9, 30, 1598, 14, 114, 16, 222, 18, 1886, 20, 5, 310, 2022, 22, 174, 15, 186, 24, 21, 60, 25, 610, 47878, 26, 13, 1378, 246, 27, 258, 28, 438, 32, 7426, 34, 1162, 36, 33, 90, 1166, 38, 66, 40, 582, 44, 12154, 46, 13054, 48, 39, 618, 6830

Offset: 1

Scott R. Shannon, Nov 19 2021

nonn

The majority of terms lie near a line of gradient 1, but some terms show extremely large jumps in value, e.g., in the first 30000 terms the largest value is a(25391) = 87893333254. It is likely all numbers eventually appear although this is unknown. In the same range, and other than a(3) = 3, there are twenty-one fixed points all between 626 to 856 inclusive. As the terms for n>30000 appear both above and below the line y = n it is possible more exist although this is unknown.

			a(3) = 3 as a(2)+3 = 9, gcd(9,3)>1, gcd(6,3)>1, gcd(3,3)>1 and 3 has not been used.
a(4) = 42 as a(3)+4 = 7, gcd(7,42)>1, gcd(3,42)>1, gcd(4,42)>1 and 42 has not been used.
a(5) = 470 as a(4)+5 = 47, gcd(47,470)>1, gcd(42,470)>1, gcd(5,470)>1 and 470 has not been used.
a(6) = 2 as a(5)+6 = 476, gcd(476,2)>1, gcd(470,2)>1, gcd(6,2)>1 and 2 has not been used.

Scott R. Shannon, Image for the first 10000 points with values below 20000. The green line is y = n.
Scott R. Shannon, Image of the first 30000 terms. On this scale the vast majority of terms lie along the x-axis.

Cf. A119018, A064413, A349472.

a[1]=1; a[2]=6; a[n_]:=a[n]=(k=2;While[MemberQ[Array[a,n-1],k]||GCD[a[n-1]+n,k]<=1||GCD[a[n-1],k]<=1||GCD[n,k]<=1,k++];k); Array[a,65] (* Giorgos Kalogeropoulos, Nov 20 2021 *)

from math import gcd
terms, appears = [1, 6], {6:True}
for n in range(3, 100):
    t = 2
    while not (appears.get(t) is None and gcd(terms[-1]+n, t)>1 and gcd(terms[-1], t)>1 and gcd(n, t)>1):
        t += 1
    appears[t] = True; terms.append(t)
print(terms) # Gleb Ivanov, Nov 20 2021

A357595 Lexicographically earliest infinite sequence of distinct positive integers such that a(n+1) is the least k != j, for which gcd(k, j) > 1; j = n + a(n).

Original entry on oeis.org

1, 4, 2, 10, 6, 22, 7, 8, 12, 3, 26, 74, 14, 9, 46, 122, 15, 16, 17, 18, 19, 5, 21, 11, 20, 24, 25, 13, 82, 27, 30, 183, 35, 28, 31, 32, 34, 142, 33, 36, 38, 158, 40, 166, 39, 42, 44, 49, 194, 45, 50, 202, 48, 303, 51, 52, 54, 37, 55, 56, 29, 57, 63, 58, 60, 65

Offset: 1

David James Sycamore, Oct 05 2022

nonn

If n + a(n) = prime p, a(n+1) is the smallest multiple (>1) of p, which has not occurred earlier. Conjectured to be a permutation of the positive integers.

			a(1)=1, then 1+a(1)=2 so a(2) must be 4, the least k != 2 which shares a divisor with 2.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14, labeling records in red and local minima in blue, highlighting primes in green and (composite) prime powers in gold.

Cf. A347113, A349472.

nn = 66; c[] = False; u = 2; a[1] = j = 1; c[1] = True; Do[Set[{k, m}, {u, n + j - 1}]; While[Or[c[k], k == m, CoprimeQ[k, m]], k++]; Set[{a[n], c[k], j}, {k, True, k}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Oct 05 2022 *)

A349491 a(1)=1, a(2)=4; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1)*n,a(n)) > 1, where a(n) != a(n-1) and a(n) != n.

Original entry on oeis.org

1, 4, 2, 6, 3, 8, 10, 5, 12, 9, 15, 14, 7, 16, 18, 20, 22, 11, 33, 21, 24, 26, 13, 27, 30, 25, 35, 32, 28, 34, 17, 36, 38, 19, 40, 39, 42, 44, 45, 46, 23, 48, 50, 52, 51, 54, 56, 49, 63, 55, 57, 58, 29, 60, 62, 31, 66, 64, 68, 65, 70, 72, 69, 74, 37, 75, 78, 76, 80, 77, 84, 81, 87

Offset: 1

Scott R. Shannon, Nov 19 2021

nonn

This sequence shows similar behavior to the EKG sequence A064413. See the linked image.

			a(3) = 2 as a(2)*3 = 6, 2!=4, 2!=3, 2 is unused and gcd(6,2) > 1.
a(4) = 6 as a(3)*4 = 8, 6!=2, 6!=4, 6 is unused and gcd(8,6) > 1.

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

Cf. A064413, A349493, A349472, A098550, A336957.

from math import gcd
terms, appears = [1], {}
for n in range(2, 100):
    t = 2
    while not(appears.get(t) is None and gcd(terms[-1]*n, t)>1 and t!=terms[-1] and t!=n):
        t += 1
    appears[t] = True; terms.append(t);
print(terms) # Gleb Ivanov, Nov 20 2021

A357614 Lexicographically earliest infinite sequence of distinct positive integers such that a(n+1) is the least k != j, for which gcd(k, j) > 1, where j = a(n) + prime(n).

Original entry on oeis.org

1, 6, 3, 2, 12, 46, 118, 5, 4, 9, 8, 13, 10, 15, 14, 122, 7, 11, 16, 166, 18, 21, 20, 206, 25, 22, 24, 254, 19, 26, 278, 27, 28, 30, 39, 32, 33, 34, 394, 17, 35, 36, 31, 37, 23, 38, 42, 44, 45, 40, 538, 48, 41, 47, 50, 614, 1754, 49, 52, 56, 674, 29, 54, 57, 58

Offset: 1

David James Sycamore, Oct 06 2022

nonn

If a(n) + prime(n) is a prime p, then a(n+1) is the smallest multiple (>1) of p that has not occurred earlier. Conjectured to be a permutation of the positive integers.

			a(1)=1; 1 + prime(1)=3, so a(2) = 6, the smallest unused number sharing a divisor with 3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, labeling maxima in red and local minima in blue, highlighting primes in green and other prime powers in gold.

Cf. A347113, A349472, A357595.

nn = 65; c[] = False; u = 2; a[1] = j = 1; c[1] = True; Do[m = j + Prime[n - 1]; If[PrimeQ[m], k = 2; While[c[k m], k++]; k *= m, k = u; While[Or[c[k], k == m, CoprimeQ[k, m]], k++]]; Set[{a[n], c[k], j}, {k, True, k}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Oct 06 2022 *)

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A349493 a(1)=1, a(2)=2; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-2)+a(n-1), a(n)) > 1 while gcd(a(n-2), a(n)) = 1 and gcd(a(n-1), a(n)) = 1.

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A349492 a(1)=1, a(2)=6; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1)+n, a(n)) > 1, gcd(a(n-1), a(n)) > 1, and gcd(n, a(n)) > 1.

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A357595 Lexicographically earliest infinite sequence of distinct positive integers such that a(n+1) is the least k != j, for which gcd(k, j) > 1; j = n + a(n).

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A349491 a(1)=1, a(2)=4; for n > 2, a(n) is the smallest unused positive number such that gcd(a(n-1)*n,a(n)) > 1, where a(n) != a(n-1) and a(n) != n.

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A357614 Lexicographically earliest infinite sequence of distinct positive integers such that a(n+1) is the least k != j, for which gcd(k, j) > 1, where j = a(n) + prime(n).

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