

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(n1) but not with a(n2).


37



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
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OFFSET

1,2


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(n1)) is nonempty,
(ii) Ker(m) intersect Ker(a(n2)) is empty, and
(iii) The set Ker(m) \ Ker(a(n1)) is nonempty.
Conjecture: The sequence is a permutation of C.


LINKS

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).


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[n1]) > 1 and igcd(k, A[n2]) = 1 then
if nops(factorset(k) minus factorset(A[n1])) > 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..n1)];


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}];


CROSSREFS

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


KEYWORD

nonn


AUTHOR



STATUS

approved



