A369361 -id:A369361 - cp's OEIS frontend

A367268 Numbers k that are not squarefree, not prime powers, and not products of primorials.

18, 20, 28, 40, 44, 45, 50, 52, 54, 56, 63, 68, 75, 76, 80, 84, 88, 90, 92, 98, 99, 100, 104, 108, 112, 116, 117, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 196, 198, 200, 204, 207, 208, 212

Offset: 1

Michael De Vlieger, Jan 28 2024

nonn

Proper subset of A369361.

Superset of A368089.

			Let P(n) = A002110(n).
12 = 6 * 2 = P(2) * P(1) is not in this sequence.
a(1) = 18 = 6 * 3 is not a product of primorials.
a(2) = 20 = 2 * 2 * 5 is not a product of primorials, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000961, A002110, A005117, A013929, A024619, A025487, A126706, A368089, A369361.

Select[Select[Range[5000], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], Nand[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1}, AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@ {#, FactorInteger[#]} &]

{a(n)} = A126706 \ A025487.

A369419 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.

Original entry on oeis.org

18, 90, 150, 630, 1050, 1470, 1890, 2100, 6930, 11550, 16170, 20790, 23100, 25410, 90090, 150150, 210210, 270270, 300300, 330330, 390390, 420420, 450450, 1531530, 2552550, 3573570, 4594590, 5105100, 5615610, 6636630, 7147140, 7657650, 8678670, 9189180, 29099070

Offset: 1

Michael De Vlieger, Mar 10 2024

nonn

			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.
n\k    3       5       7       9      10      11
------------------------------------------------
2:    18;
3:    90,    150;
4:   630,   1050,   1470,   1890,   2100;
5:  6930,  11550,  16170,  20790,  23100,  25410;
    ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

P = 2; nn = 8;
s = Select[Range[3, Prime[nn+1]],
  Nor[IntegerQ@ Log2[#],
      And[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1},
          AllTrue[Differences@ #2[[All, -1]], # <= 0 &]]] & @@
    {#, FactorInteger[#]} &];
Table[P *= Prime[n]; P*TakeWhile[s, # < Prime[n + 1] &], {n, 2, nn}]

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

Intersection of A364998 and A056808.

cp's OEIS Frontend

A367268 Numbers k that are not squarefree, not prime powers, and not products of primorials.

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