A369954 - OEIS

%I #30 Apr 04 2024 10:01:54

%S 175,245,275,325,425,475,539,575,605,637,725,775,833,845,847,875,925,

%T 931,1025,1075,1127,1175,1183,1225,1325,1375,1421,1445,1475,1519,1525,

%U 1573,1625,1675,1715,1775,1805,1813,1825,1859,1925,1975,2009,2023,2057,2075,2107

%N Numbers k that are neither squarefree nor prime powers and also coprime to 6.

%C Define quality Q to signify a number k neither squarefree nor prime power, i.e., k is in A126706. For example, 12 has quality Q but numbers k = 1..11 do not.

%C Numbers k in this sequence have quality Q and are such that either (k-1) or (k+1) also have quality Q. Hence k also appears in A369276, but not in A369516.

%C Numbers k such that k mod 12 = 1 or k mod 12 = 5 imply (k-1) in A126706, since 4 divides (k-1).

%C Numbers k such that k mod 12 = 7 or k mod 12 = 11 imply (k+1) in A126706, since 4 divides (k+1).

%C Proper subset of A367455.

%C By definition these odd numbers are such that A053669(k) = 2, therefore A053669(k) < A003557(k), hence this sequence is a proper subset of A360765.

%H Michael De Vlieger, <a href="/A369954/b369954.txt">Table of n, a(n) for n = 1..10000</a>

%F Intersection of A007310 and A126706.

%F Intersection of A007310, A013929, and A024619.

%t Select[Flatten[Array[6 # + {1, 5} &, 360]], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]

%o (PARI) isok(k) = !issquarefree(k) && !isprimepower(k) && (gcd(k, 6)==1); \\ _Michel Marcus_, Mar 25 2024

%Y Cf. A003557, A007310, A008586, A013929, A024619, A053669, A126706, A356322, A369276, A360765, A367455, A369516.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Mar 24 2024