cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A372975 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1) while omega(a(n)) does not equal omega(a(n-1)).

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 10, 5, 15, 9, 18, 8, 14, 7, 21, 27, 24, 16, 20, 25, 30, 22, 11, 33, 42, 26, 13, 39, 60, 28, 32, 34, 17, 51, 66, 36, 64, 38, 19, 57, 78, 40, 70, 35, 49, 56, 84, 44, 90, 45, 81, 48, 102, 46, 23, 69, 105, 50, 110, 52, 114, 54, 120, 55, 121, 77, 126, 58, 29, 87, 132, 62, 31, 93, 138
Offset: 1

Views

Author

Scott R. Shannon, May 26 2024

Keywords

Comments

The sequence shows similar behavior to the EKG sequence A064413; for the terms studied the primes appear in the natural order, and when a prime p is a term, the proceeding and following terms are 2p and 3p respectively.
For larger n a graph of the sequence also displays very similar behavior to A064413, although for the first ~2500 terms the main concentration of terms are along two lines which eventually join - see the attached image of the first 5000 terms.
The fixed points begin 1, 2, 22, 26, 36, 38, 1991, 2023, 2159, 2189, 2627; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.
From Michael De Vlieger, May 28 2024: (Start)
Four general trajectories become apparent in log log scatterplot:
1. Beta, the trajectory of primes a(j) = p.
2. Alpha, the trajectory of numbers a(j+1) = 3*p.
3. Delta, the trajectory of perfect prime powers and numbers k with omega(k) = 3.
4. Gamma, the trajectory of all other (composite) numbers.
Delta begins with a(21) = 30 and merges with gamma around n = 2958. The merger alters the "slope" of all trajectories as a result. Thereafter, a number k with omega(k) = 2 is comparable in size with one that has omega(k) = 3. This does not seem to happen for omega(k) = 4, etc. (See a(2102) = 2310).
Perfect prime powers may technically constitute a separate, more scattered trajectory superposed upon delta. Still, the merger with gamma seems to occur around the same point as with delta.
Exception to first comment: 12 follows 3, since omega(9) = omega(3). The number 12 lies outside trajectory alpha, since 12 = 4*3. (End)

Examples

			a(3) = 6 as a(2) = 2 and omega(2) = A001221(2) = 1, and 6 shares a factor with 2 while omega(6) = A001221(6) = 2 which does not equal 1.
		

Crossrefs

Programs

  • Mathematica
    nn = 1000; c[] := False; m[] := 1;
    Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; v = 4;
    Do[Which[
      And[PrimeQ[#], OddQ[#]] &[j/2], k = j/2,
      PrimePowerQ[j], k = FactorInteger[j][[1, 1]];
        While[Or[c[#], PrimePowerQ[#]] &[k*m[k]], m[k]++]; k *= m[k],
      True, k = v;
        While[Or[CoprimeQ[j, k], PrimeNu[k] == #, c[k]] &[PrimeNu[j]], k++]];
      Set[{a[n], c[k], j}, {k, True, k}];
      If[k == v, While[Or[PrimeQ[v], c[v]], v++]], {n, 3, nn}];
    Array[a, nn] (* Michael De Vlieger, May 28 2024 *)

A353917 a(1) = 4. Let j = a(n-1) and let m = omega(gcd(j, k)) with gcd(j, k) > 1. For n > 1, a(n) = least k such that min(omega(j), omega(k)) = m and m < max(omega(j), omega(k)), but neither j | k nor k | j.

Original entry on oeis.org

4, 6, 8, 10, 16, 12, 9, 15, 25, 20, 30, 18, 27, 21, 49, 14, 32, 22, 64, 24, 42, 28, 70, 40, 60, 36, 66, 44, 110, 50, 90, 48, 78, 52, 128, 26, 169, 39, 81, 33, 121, 55, 125, 35, 343, 56, 84, 54, 102, 68, 170, 80, 120, 45, 105, 63, 168, 72, 114, 76, 190, 100, 130
Offset: 1

Views

Author

Michael De Vlieger, May 10 2022

Keywords

Comments

The sequence exhibits phases involving alternating composite prime powers and squarefree semiprimes. These manifest in log-log scatterplot in a caustic fashion, where the composite prime power is very much larger than the squarefree semiprime for sufficiently large n.
Let P = the set of distinct prime divisors of j = a(n-1), and let Q = the set of distinct prime divisors of k = a(n). Let g = gcd(j, k) > 1 and let G = {P intersect Q}. Noncoprime j and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| == |G|, or vice versa, yet neither j | k nor k | j.
Theorem: primes are prohibited. Proof: since we have gcd(j, k) > 1 and do not allow divisibility, and since primes must either divide or be coprime to another number m, primes do not appear in this sequence.
Theorem: squarefree semiprimes j = pq are followed by k = p^2 or k = q^2. Proof: since omega(j) = |P| = 2 and is squarefree, we have 2 cases pertaining to successor k, both with gcd(j, k) > 1.
1.) |P| == |G| implies |Q| > |G| and |Q| > |P|.
2.) |Q| == |G| implies |P| > |G| and |P| > |Q|.
The first case implies some prime r | k yet gcd(j, r) = 1. But this would require j | k, which is prohibited. The second case suggests either p | k or q | k, but so as to satisfy non-divisibility axiom, we are forced into either k = p^e, e > 1, or k = q^m, m > 1.
Corollary: powers of the same prime appear in natural order in this sequence.
There is a weaker alternation between numbers in A120944 and A350352 as n is sufficiently large. This alternation exhibits prime power factor features akin to the composite prime power-squarefree semiprime alternation.
Conjecture: permutation of composite numbers.

Crossrefs

Programs

  • Mathematica
    nn = 2^7; c[_] = 0; j = a[1] = 4; c[4] = 1; u = 6; Do[Set[k, u]; Set[m, PrimeNu[j]]; While[Nand[c[k] == 0, ! Divisible[#2, #1] & @@ Sort[{j, k}], And[#2 > #3, #1 == #3] & @@ Append[Sort[{m, PrimeNu[k]}], PrimeNu[GCD[j, k]]]], k++]; Set[{a[i], c[k]}, {k, i}]; j = k; If[k == u, While[Nand[c[u] == 0, CompositeQ@ u], u++]], {i, 2, nn}]; Array[a, nn]

A354720 a(n) = n for n <= 3; let i = a(n-2) and j = a(n-1); a(n+1) = least k not already in the sequence such that (j, k) = 1 and (i, k) = m > 1 and only one of either omega(i) or omega(k) exceed omega(m), where omega = A001221.

Original entry on oeis.org

1, 2, 3, 10, 21, 4, 7, 6, 35, 8, 5, 12, 55, 9, 11, 15, 22, 25, 16, 45, 14, 27, 32, 33, 20, 81, 64, 39, 28, 13, 42, 65, 18, 125, 66, 85, 24, 17, 30, 119, 36, 49, 60, 77, 40, 121, 70, 99, 50, 231, 128, 63, 26, 105, 169, 75, 52, 165, 182, 135, 56, 195, 154, 117, 44
Offset: 1

Views

Author

Michael De Vlieger, Jun 23 2022

Keywords

Comments

Theorem: even terms cannot be adjacent. Proof: If prime p | j, then p cannot divide k as well, because then (j, k) >= p and by definition of "prime", p > 1, which contradicts the axiom (j, k) = 1. Since 2 is prime, consecutive even terms are prohibited.
A restriction on the Yellowstone sequence A098550 analogous to A353916 regarding its relationship to A064413.

Crossrefs

Programs

  • Mathematica
    nn = 120; s = Range[3]; state = {2, 3, 4, 7}; u = 1; c[] = 0; f[j, k_] := Which[j == k, 5, GCD[j, k] == 1, 0, True, 1 + FromDigits[Map[Which[Mod[##] == 0, 1, PowerMod[#1, #2, #2] == 0, 2, True, 0] & @@ # &, Permutations[{k, j}]], 3]]; Array[Set[{a[s[[#]]], c[#]}, {#, s[[#]]}] &, Length[s]]; While[c[u] > 0, u++]; Set[{i, j}, s[[-2 ;; -1]]]; Do[k = u; While[Nand[c[k] == 0, MemberQ[state, f[i, k]], CoprimeQ[j, k]], k++]; Set[{a[n], c[k], i, j}, {k, n, j, k}]; If[k == u, While[c[u] > 0, u++]], {n, Length[s] + 1, nn}]; Array[a, nn]
Showing 1-3 of 3 results.