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Numbers k that are neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is not a primorial.
2

%I #29 Apr 04 2024 10:01:12

%S 18,90,150,630,1050,1470,1890,2100,6930,11550,16170,20790,23100,25410,

%T 90090,150150,210210,270270,300300,330330,390390,420420,450450,

%U 1531530,2552550,3573570,4594590,5105100,5615610,6636630,7147140,7657650,8678670,9189180,29099070

%N Numbers k that are neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is not a primorial.

%H Michael De Vlieger, <a href="/A369419/b369419.txt">Table of n, a(n) for n = 1..10000</a>

%F This sequence is { k = m*P(i) : 3 <= m < prime(i), i > 1, m in A369361 }.

%F Intersection of A364998 and A056808.

%e Seen as an irregular triangle T(n,k) of rows n where T(n,k) = P(n)*k, and k < prime(n+1) is in A369361.

%e n\k 3 5 7 9 10 11

%e ------------------------------------------------

%e 2: 18;

%e 3: 90, 150;

%e 4: 630, 1050, 1470, 1890, 2100;

%e 5: 6930, 11550, 16170, 20790, 23100, 25410;

%e ...

%t P = 2; nn = 8;

%t s = Select[Range[3, Prime[nn+1]],

%t Nor[IntegerQ@ Log2[#],

%t And[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1},

%t AllTrue[Differences@ #2[[All, -1]], # <= 0 &]]] & @@

%t {#, FactorInteger[#]} &];

%t Table[P *= Prime[n]; P*TakeWhile[s, # < Prime[n + 1] &], {n, 2, nn}]

%Y Cf. A002110, A003557, A007947, A025487, A053669, A055932, A056808, A060735, A119288, A364998, A369361, A369540, A369541.

%K nonn

%O 1,1

%A _Michael De Vlieger_, Mar 10 2024