A109465 -id:A109465 - cp's OEIS frontend

A241012 Inverse permutation to A109465.

1, 2, 4, 6, 8, 3, 10, 12, 14, 5, 16, 7, 18, 9, 11, 20, 22, 13, 24, 15, 17, 19, 26, 21, 28, 23, 30, 25, 32, 27, 34, 36, 29, 31, 33, 35, 38, 37, 39, 41, 40, 42, 43, 44, 46, 48, 45, 50, 47, 52, 54, 56, 49, 58, 60, 62, 64, 66, 51, 53, 55, 68, 70, 57, 72, 59, 61

Offset: 1

Reinhard Zumkeller, Aug 07 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

import Data.List (elemIndex); import Data.Maybe (fromJust)
a241012 = (+ 1) . fromJust . (`elemIndex` a109465_list)

A143691 Lexicographically earliest permutation of the natural numbers having alternating even and odd numbers of prime factors.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 9, 7, 10, 8, 14, 11, 15, 12, 16, 13, 21, 17, 22, 18, 24, 19, 25, 20, 26, 23, 33, 27, 34, 28, 35, 29, 36, 30, 38, 31, 39, 32, 40, 37, 46, 41, 49, 42, 51, 43, 54, 44, 55, 45, 56, 47, 57, 48, 58, 50, 60, 52, 62, 53, 64, 59, 65, 61, 69, 63, 74, 66, 77, 67, 81, 68

Offset: 1

Reinhard Zumkeller, Aug 29 2008

nonn

A066829(a(n)*a(n+1)) = 1; n mod 2 + A001222(a(n)) = 1;
a(2*n) = A026424(n); a(2*n-1) = A028260(n);
a(a(n)) = A143693(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A143692 (inverse), A001222, A066829, A026424, A028260, A143693, A109465.

import Data.List (delete)
a143691 n = a143691_list !! (n-1)
a143691_list = f 1 [1..] where
   f m xs = g xs where
     g (z:zs) = if m + m' /= 1 then g zs else z : f m' (delete z xs)
                where m' = a001222 z `mod` 2
-- Reinhard Zumkeller, Aug 07 2014

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

Scott R. Shannon, May 26 2024

nonn

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)

			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.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 5000 terms. The green line is a(n) = n.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, 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. A001221, A027748, A372974, A064413, A352588, A109465, A353916, A372974.

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 *)

A372974 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and omega(a(n)) does not equal omega(a(n-1)).

Original entry on oeis.org

1, 2, 15, 4, 21, 5, 6, 7, 10, 3, 14, 9, 20, 11, 12, 13, 18, 17, 22, 19, 24, 23, 26, 25, 28, 27, 34, 29, 30, 31, 33, 8, 35, 16, 39, 32, 45, 37, 36, 41, 38, 43, 40, 47, 42, 53, 44, 49, 46, 59, 48, 61, 50, 67, 51, 64, 55, 71, 52, 73, 54, 79, 56, 81, 58, 83, 57, 70, 69, 89, 60, 77, 78, 85, 66, 65, 84

Offset: 1

Scott R. Shannon, May 26 2024

nonn

The fixed points show an unusual pattern; they begin 1, 2, 4, 69, 190, 438, 545, 725, 732, 909 and it appears, based on a graph of the sequence (see the attached image of the first 5000 terms) there may be no more. However more exist at 324388, 330574, 333069, 333531,..., 369752. Then once again there is a large gap until 2704713, 2726054, 2760963, ... . It is unclear what causes this behavior.

The sequence is conjectured to be a permutation of the positive integers.

			a(3) = 15 as a(2) = 2 and omega(2) = A001221(2) = 1, and 15 is coprime to 2 while omega(15) = A001221(15) = 2 which does not equal 1. No smaller number satisfies both of these requirements.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 5000 terms. The green line is a(n) = n.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, with purple indicating powerful numbers that are not prime powers.

Cf. A001221, A027748, A372975, A352588, A109465.

nn = 120; c[_] := False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; u = 3;
Do[k = u;
  While[Or[GCD[j, k] > 1, PrimeNu[k] == #, c[k]] &[PrimeNu[j]], k++];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == u, While[c[u], u++]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, May 28 2024 *)

A373350 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

Original entry on oeis.org

1, 2, 6, 30, 3, 10, 42, 4, 12, 60, 5, 14, 66, 7, 15, 70, 8, 18, 78, 9, 20, 84, 11, 21, 90, 13, 22, 102, 16, 24, 105, 17, 26, 110, 19, 28, 114, 23, 33, 120, 25, 34, 126, 27, 35, 130, 29, 36, 132, 31, 38, 138, 32, 39, 140, 37, 40, 150, 41, 44, 154, 43, 45, 156, 47, 46, 165, 49, 48, 168, 53, 50, 170, 59

Offset: 1

Scott R. Shannon, Jun 01 2024

nonn

The terms of the sequence are initially concentrated along three lines, each line containing terms with either one, two, or three-or-more distinct prime factors. See the attached images. The two lower lines, containing terms with one and two distinct prime factors, cross near n = 66, and then the upper two, containing terms with one and three-or-more distinct prime factors, cross near n = 326. At this point the upper line changes and contains terms with one distinct prime factor but also terms with more than three distinct prime factors. After a large gap the lower two lines, containing terms with two and three distinct prime factors, cross near n = 25308, and then finally, after an even longer gap, the upper two lines, containing all terms except those with three distinct prime factors, merge and become one line near n = 344310. It is likely all subsequent terms fall into one of the two remaining lines, although this is unknown.
A number with one distinct prime factor appears as a term for all n with n mod 3 = 2 up until n = 350, when a(350) = 510 = 2*3*5*17 is smaller than the next unused number with one distinct prime factor. Likewise a number with three distinct prime factors appears as a term for all n with n mod 3 = 1 up until n = 100 when a(100) = 210 = 2*3*5*7 is smaller than the next unused number with three distinct prime factors. However, rather surprisingly, a number with two distinct prime factors appears as a term for all n with n mod 3 = 0 all the way up until n = 344307, when a(344307) = 373215 = 3*5*139*179 is smaller than the next unused number with two distinct prime factors. This value of n corresponds to the start of the merging of the upper two lines near n = 344310 described above.
The fixed points begin 1, 2, 15, 18, 125, 137, 2737, 120051; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(4) = 30 as a(2) = 2 has one distinct prime factor and a(3) = 6 has two distinct prime factors, and 30 is the smallest unused number with three distinct prime factors.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 500 terms. In this and other images terms with one, two, three, or more than three distinct prime factors are shown in red, yellow, green, violet respectively. The white line is a(n) = n.
Scott R. Shannon, Image of the first 50000 terms.
Scott R. Shannon, Image of the first 400000 terms.

Cf. A001221, A027748, A109465, A372975.

cp's OEIS Frontend

A241012 Inverse permutation to A109465.

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A143691 Lexicographically earliest permutation of the natural numbers having alternating even and odd numbers of prime factors.

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Haskell

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)).

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Mathematica

A372974 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) is coprime to a(n-1) and omega(a(n)) does not equal omega(a(n-1)).

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Mathematica

A373350 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that omega(a(n)) does not equal omega(a(n-1)) or omega(a(n-2)).

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