A358093 -id:A358093 - cp's OEIS frontend

A358534 Start with a(1)=1, a(2)=2. Thereafter, if gcd(a(n-2),a(n-1)) = 1 then a(n) is the smallest unused k such that gcd(a(n-2),k) > 1 and gcd(a(n-1),k) = 1, otherwise a(n) is the smallest unused k such that gcd(a(n-2),k) = 1 and gcd(a(n-1),k) > 1. If the latter is impossible, then a(n) = smallest missing number u. (See comments.)

1, 2, 4, 3, 9, 5, 10, 6, 25, 35, 7, 15, 12, 55, 11, 20, 8, 45, 21, 40, 16, 65, 13, 30, 14, 27, 33, 17, 34, 18, 85, 95, 51, 24, 119, 49, 68, 22, 153, 39, 136, 28, 187, 99, 170, 26, 75, 57, 50, 32, 105, 63, 80, 38, 115, 23, 60, 36, 125, 145, 19, 76, 29, 87, 31, 62

Offset: 1

Michael De Vlieger and David James Sycamore, Nov 20 2022

A lexicographically earliest sequence. We write (i,j) to mean gcd(i,j) for brevity. Let rad(m) = A007947(m).
Define (i,j)=1 to be "closed" else "open". If we have i and j closed, then we find the least k not already in the sequence that is closed to i but open to j, otherwise we find same that is open to i but closed to j.
Condition [1], (i,j)=1 requires smallest unused k such that (i,k)=1 and (j,k)>1, given primes p|i, merely by finding the smallest missing k indivisible by any p.
Condition [2], (i,j)>1, requires smallest unused k such that (i,k)>1 and (j,k)=1 iff rad(i) does not divide rad(j).
Let S = {p | i} and T = {q | j}. If T contains S, then (j,k)=1 makes (i,k)>1 impossible.
Condition [3] allows a solution. In this case, we simply set a(n) = u, the smallest missing number in a(1..n-1).
The most conspicuous case of [3] is i = p^v and j = p^w, v < w. Since p^v | p^w, if a(n) = k is made such that (p^w,k)=1, then k is also coprime to p^v or any power of p. Hence we write a(n) = u.
Another, rare case of [3] is if i and j belong to R, the sequence of numbers that are products of the same squarefree kernel K. Suppose K = 6, then R = A003586. Now suppose i = 6*R(2) = 6*2 = 12 and j = 6*R(3) = 6*3 = 18. Indeed, if a(n) = k is made such that (18,k)=1, then k is also coprime to 12 and any number in 6*R. Hence in this case we write a(n) = u.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing records in red, local minima in blue, highlighting primes in green and other prime powers in gold.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing terms generated by condition [1] in blue, [2] in green, and [3] in red.

Cf. A358093.

nn = 120; c[] = False; q[] = 1; f[n_] := Times @@ FactorInteger[n][[All, 1]]; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; Set[{i, j, u}, {a[1], a[2], 3}]; Set[{r, s}, {f[i], f[j]}]; Do[Which[CoprimeQ[i, j], If[PrimeNu[j] == 1, While[c[q[s] s], q[s]++]; k = q[s]; While[Nand[! c[s k], CoprimeQ[i, k]], k++]; k *= s, k = u; While[Nand[! c[k], CoprimeQ[i, k], ! CoprimeQ[j, k]], k++]], Divisible[s, r], k = u, True, If[PrimeNu[i] == 1, While[c[q[r] s], q[r]++]; k = q[r]; While[Nand[! c[r k], CoprimeQ[j, k]], k++]; k *= r, k = u; While[Nand[! c[k], ! CoprimeQ[i, k], CoprimeQ[j, k]], k++]] ]; Set[{a[n], c[k], i, j, r, s}, {k, True, j, k, s, f[k]}]; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn]

A358354 a(n) = n for n <= 3. Thereafter a(n) is the least m such that rad(m) = rad(rad(a(n-3)) + rad(a(n-1))) where rad is A007947.

Original entry on oeis.org

1, 2, 3, 4, 8, 5, 7, 9, 16, 27, 6, 32, 25, 11, 13, 12, 17, 30, 18, 23, 53, 59, 82, 15, 74, 78, 93, 167, 35, 64, 169, 24, 128, 45, 21, 529, 38, 3481, 164, 60, 89, 57, 87, 22, 79, 166, 94, 173, 339, 433, 606, 105, 538, 286, 391, 929, 75, 406, 1335, 90, 218, 1553

Offset: 1

David James Sycamore, Nov 11 2022

nonn

Similar sequence to A358093, though here the definition works between a(n-1) and a(n-3) (skipping a(n-2)) so there is no evident parity pattern, and adjacent pairs are not always coprime.

Conjecture: for n > 60, a(n) is squarefree. - Michael De Vlieger, Jul 01 2025

			To find a(4): a(1)=1, a(3)=3 so a(4) is the least m such that rad(n) = rad(rad(1) + rad(3)) = rad(4) = 2. Since 2 has occurred already, a(4) = 4.

Michael De Vlieger, Table of n, a(n) for n = 1..1000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..256, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple represents powerful numbers that are not prime powers.

Cf. A007947, A358093.

Block[{a, c, f, i, j, k, m, p, q, s, nn}, nn = 62; c[] = False; p[] = q[] = 1; f[n] := Times @@ FactorInteger[n][[All, 1]]; Array[Set[{a[#], c[#]}, {#, True}] &, 3]; Array[(q[#]++; p[#]++) &[f[a[#]] ] &, 3]; Set[{i, j, k}, Array[f[a[#]] &, 3]]; Do[s = f[i + k]; If[PrimeQ[s], m = s^p[s]; p[s]++, m = q[s]; While[Nand[! c[m s], PowerMod[s, s, m] == 0], m++]; m *= s]; q[s]++; Set[{a[n], c[m], i, j, k}, {m, True, j, k, s}], {n, 4, nn}]; Array[a, nn] ] (* Michael De Vlieger, Nov 12 2022 *)

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