A369276 -id:A369276

Search: a369276 -id:a369276

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A370372

Row lengths of A369276.

+20
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. A126706, A369276.

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, Mar 24 2024

STATUS

approved

A369516

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

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

EXAMPLE

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.

MATHEMATICA

Select[Select[Range[264], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], NoneTrue[{# - 1, # + 1}, Nor[SquareFreeQ[#], PrimePowerQ[#]] &] &]

CROSSREFS

Cf. A126706, A369276, A369954.

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, Mar 24 2024

STATUS

approved

A369954

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

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

FORMULA

Intersection of A007310 and A126706.

Intersection of A007310, A013929, and A024619.

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, Mar 24 2024

STATUS

approved

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