A369276 -id:A369276 - cp's OEIS frontend

A370372 Row lengths of A369276.

2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Michael De Vlieger, Mar 24 2024

Also 1 more than the number of consecutive 1s in the n-th occasion of a run of 1s in A358089.

			Define quality Q to signify a number neither squarefree nor prime power. For example, 12 has quality Q but smaller numbers do not.
The smallest number k with quality Q such that either (k-1) or (k+1) (or both) share quality Q is 44.
Since both {44, 45} have quality Q, but 43 and 46 are squarefree, a(1) = 2.
Since both {75, 76} have quality Q, but 74 and 78 are squarefree, a(2) = 2.
Since all of {98, 99, 100} have quality Q but 97 and 101 are prime, a(3) = 3, etc.

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

Cf. A126706, A369276.

1 + Map[Length, SplitBy[Differences@ Select[Range[1000], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], # == 1 &]][[2 ;; -1 ;; 2]]

A369516 Numbers k in A126706 such that neither k-1 nor k+1 is in A126706.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 48, 50, 52, 54, 56, 60, 63, 68, 72, 80, 84, 88, 90, 92, 96, 104, 108, 112, 120, 124, 126, 132, 140, 144, 150, 156, 160, 162, 164, 168, 180, 184, 192, 196, 198, 200, 204, 212, 216, 220, 228, 232, 234, 236, 240, 242, 248, 250, 252, 264

Offset: 1

Michael De Vlieger, Mar 24 2024

Singletons in A126706.
The smallest odd term is 63.
Terms are even or divisible by 3, or both. Does not include k coprime to 6; k in A369954 are not in this sequence.

			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 k = 1..11 do not.
The number 12 is in the sequence since it has quality Q, but neither 11 nor 13 do.
The number 44 is not in the sequence since 45 has quality Q.
The number 99 is not in the sequence because both 98 and 100 have quality Q, etc.

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

Cf. A126706, A369276, A369954.

Select[Select[Range[264], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], NoneTrue[{# - 1, # + 1}, Nor[SquareFreeQ[#], PrimePowerQ[#]] &] &]
Mean/@SequencePosition[Table[If[!SquareFreeQ[n]&&!PrimePowerQ[n],1,0],{n,300}],{0,1,0}] (* Harvey P. Dale, Jan 30 2025 *)

A369954 Numbers k that are neither squarefree nor prime powers and also coprime to 6.

Original entry on oeis.org

175, 245, 275, 325, 425, 475, 539, 575, 605, 637, 725, 775, 833, 845, 847, 875, 925, 931, 1025, 1075, 1127, 1175, 1183, 1225, 1325, 1375, 1421, 1445, 1475, 1519, 1525, 1573, 1625, 1675, 1715, 1775, 1805, 1813, 1825, 1859, 1925, 1975, 2009, 2023, 2057, 2075, 2107

Offset: 1

Michael De Vlieger, Mar 24 2024

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.
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.
Numbers k such that k mod 12 = 1 or k mod 12 = 5 imply (k-1) in A126706, since 4 divides (k-1).
Numbers k such that k mod 12 = 7 or k mod 12 = 11 imply (k+1) in A126706, since 4 divides (k+1).
Proper subset of A367455.
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.

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

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

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

isok(k) = !issquarefree(k) && !isprimepower(k) && (gcd(k, 6)==1); \\ Michel Marcus, Mar 25 2024

Intersection of A007310 and A126706.

Intersection of A007310, A013929, and A024619.

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A370372 Row lengths of A369276.

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A369516 Numbers k in A126706 such that neither k-1 nor k+1 is in A126706.

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A369954 Numbers k that are neither squarefree nor prime powers and also coprime to 6.

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