A360519 Let C consist of 1 together with all numbers with at least two distinct prime factors; this is the lexicographically earliest infinite sequence {a(n)} of distinct elements of C such that, for n>2, a(n) has a common factor with a(n-1) but not with a(n-2).

1, 6, 10, 35, 21, 12, 20, 55, 33, 18, 14, 77, 99, 15, 40, 22, 143, 39, 24, 28, 91, 65, 30, 34, 119, 63, 36, 26, 221, 51, 42, 38, 95, 45, 48, 44, 187, 85, 50, 46, 69, 57, 76, 52, 117, 75, 70, 58, 87, 93, 62, 56, 105, 111, 74, 68, 153, 123, 82, 80, 115, 161, 84, 60, 145, 203, 98, 54, 129, 215, 100, 66, 141

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Mar 01 2023

Keywords

• nonn

Comments

In other words, C contains all positive numbers except powers of primes p^k, k>=1.

This is a modified version of the Enots Wolley sequence A336957. The modification ensures that the sequence does not contain the prime 2.

Let Ker(k), the kernel of k, denote the set of primes dividing k. Thus Ker(36) = {2,3}, Ker(1) = {}.

Theorem: a(1)=1, a(2)=6; thereafter, a(n) is the smallest number m not yet in the sequence such that

(i) Ker(m) intersect Ker(a(n-1)) is nonempty,

(ii) Ker(m) intersect Ker(a(n-2)) is empty, and

(iii) The set Ker(m) \ Ker(a(n-1)) is nonempty.

Conjecture: The sequence is a permutation of C.

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..100000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing squarefree composites (in A120944) in green, numbers neither prime power nor squarefree (in A126706) in blues, highlighting powerful numbers (in A001694 and A286708) in large dark blue.
• Scott R. Shannon, White on black graph of first 10^5 terms [Some aspects of the graph are more visible when black and white are swapped. The green line is a(n) = n]
• Scott R. Shannon, Image of the first 1 million terms in color. The terms with a lowest prime factor of 2,3,5,7,9,11,13,17,19,>=23 are colored white, red, orange, yellow, green, blue, indigo, violet, gray respectively.
• N. J. A. Sloane, Table showing a(1)-a(13), also the smallest missing number (smn, A361109 and A361110), binary vectors showing which terms are divisible by the primes 2, 3, 5, 7, 11; and phi, a decimal representation of those binary vectors (A361111).

Crossrefs

Cf. A098550, A336957.

For a number of sequences related to this, see A361102 (the sequence C) and the following entries.

Programs

Maple

with(numtheory);
N:= 10^4: # to get a(1) to a(n) where a(n+1) is the first term > N
B:= Vector(N, datatype=integer[4]):
A[1]:=1; A[2]:=6;
for n from 3 do
  for k from 10 to N do
    if B[k] = 0 and igcd(k, A[n-1]) > 1 and igcd(k, A[n-2]) = 1 then
          if nops(factorset(k) minus factorset(A[n-1])) > 0 then
       A[n]:= k;
       B[k]:= 1;
       break;
          fi;
    fi
  od:
  if k > N then break; fi;
od:
s1:=[seq(A[i], i=1..n-1)];

Mathematica

nn = 2^12; c[_] = False;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
MapIndexed[
 Set[{a[First[#2]], c[#1]}, {#1, True}] &, {1, 6}];
 u = 10; i = a[1]; j = a[2];
Do[k = u;
  While[Nand[! PrimePowerQ[k], ! c[k],
    CoprimeQ[i, k], ! CoprimeQ[j, k], ! Divisible[j, f[k]]], k++];
  Set[{a[n], c[k], i, j}, {k, True, j, f[k]}];
  If[k == u, While[Or[c[u], PrimePowerQ[u]], u++]]
  , {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Mar 03 2023 *)