OFFSET
1,1
COMMENTS
Like A280864, this sequence can be partitioned into blocks on the occasion of coprime a(j) and a(j+1).
This sequence eliminates the confinement of a certain class of number (wherein the class is related to prime power decomposition) to the first or last terms in a block. In s = A280864, for instance, prime powers may only occur at the beginning or end of a block, that is, such that s(n) is coprime to at least one of s(n-1) or s(n+1).
This said, this sequence preserves the following qualities pertinent to A280864:
A.) Terms a(j) that are coprime to a neighbor (at the beginning or end of a block) are such that rad(a(j)) divides the other neighbor (where rad = A007947),
B.) Terms a(n) that are noncoprime to both neighbors (i.e., in the middle of a block) are such that rad(a(j)) divides neither neighbor.
Duplex blocks {k,m} have rad(k) = rad(m), and apart from the first, {12,18}, k | m. Are there any other duplex blocks such that rad(k) = rad(m) yet k does not divide m?
Agrees with A397127 through a(9); S20260622(10) = 36, which is forbidden in this sequence.
Conjecture: permutation of A332785.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..16384
EXAMPLE
Table of the first 16 terms:
n a(n) Prime power factors
------------------------------
1 12 = 2^2 3
2 18 = 2 3^2
3 175 = 5^2 7
4 140 = 2^2 5 7
5 24 = 2^3 3
6 45 = 3^2 5
7 20 = 2^2 5
8 28 = 2^2 7
9 63 = 3^2 7
10 48 = 2^4 3
11 40 = 2^3 5
12 75 = 3 5^2
MATHEMATICA
Block[{c, k, m, r, s, w, nn, rad},
nn = 2^15; c[_] := False; r = s = 1;
rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
Reap[Do[w = GCD[r, s]; m = r/w; k = m;
While[Or[c[k], SquareFreeQ[k],
Divisible[k, rad[k]^2], ! CoprimeQ[w, k] ], k += m];
Sow[k]; Set[{c[k], s, r}, {True, r, rad[k] } ],
{i, nn} ] ][[-1, 1]] ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jun 21 2026
STATUS
approved