A101932 -id:A101932 - cp's OEIS frontend

A076763 1-apexes of omega: numbers n such that omega(n-1) < omega(n) > omega(n+1), where omega(m) = the number of distinct prime factors of m.

6, 10, 12, 18, 24, 26, 28, 30, 42, 48, 60, 66, 70, 72, 78, 80, 82, 84, 90, 102, 105, 108, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 192, 195, 198, 204, 210, 220, 222, 228, 234, 238, 240, 242, 246, 252, 255

Offset: 1

Joseph L. Pe, Nov 13 2002

I call n a "k-apex" (or "apex of height k") of the arithmetical function f if n satisfies f(n-k) < ... < f(n-1) < f(n) > f(n+1) > .... > f(n+k).

The terms here are the positions of the positive terms in A101941. Note, however, the differences between the definition of k-apex and Neil Fernandez's definition of k-peak in A101941. - Peter Munn, May 26 2023

			28 is in the sequence because it has two unique prime factors (2 and 7), more than either of its neighbors 27 (one such factor, namely 3) and 29 (one such factor, 29). - _Neil Fernandez_, Dec 21 2004

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001221, A101932, A101941.

omega[n_] := Length[FactorInteger[n]]; Select[Range[3, 500], omega[ # - 1] < omega[ # ] > omega[ # + 1] &]
For[i=1, i<1000, If[And[Length[FactorInteger[i-1]]Neil Fernandez, Dec 21 2004 *)
#[[2,1]]&/@Select[Partition[Table[{n,PrimeNu[n]},{n,300}],3,1],#[[1,2]]<#[[2,2]]>#[[3,2]]&] (* Harvey P. Dale, Dec 11 2011 *)

isok(n) = (omega(n-1) < omega(n)) && (omega(n) > omega(n+1)); \\ Michel Marcus, May 06 2017

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

A101934 Numbers n with omega(n) smaller than omega(n-1) and omega(n+1).

Original entry on oeis.org

11, 13, 19, 23, 25, 27, 29, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 131, 137, 139, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 221, 223, 227

Offset: 1

Neil Fernandez, Dec 21 2004

			125 is in the sequence because it has one unique prime factor (5), which is fewer than its neighbors 124 (two such factors, namely 2 and 31) and 126 (two such factors, namely 2 and 53).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001221, A101932, A076763.

For[i=2, i<1000, If[And[Length[FactorInteger[i 1]]>Length[FactorInteger[i]], Length[FactorInteger[i+1]] > Length[FactorInteger[i]]], Print[i]];i++ ]
(* Second program: *)
Select[Range[1000], PrimeNu[#] < Min[PrimeNu[#-1], PrimeNu[#+1]]&] (* Jean-François Alcover, Nov 14 2016 *)
Flatten[Position[Partition[PrimeNu[Range[250]],3,1],?(#[[1]]>#[[2]]<#[[3]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Apr 18 2021 *)

A101294 Numbers n such that omega(n-2) = omega(n-1) = omega(n) = omega(n+1) = omega(n+2).

Original entry on oeis.org

56, 93, 94, 117, 143, 144, 145, 146, 160, 207, 214, 215, 216, 217, 297, 303, 325, 326, 327, 393, 537, 687, 723, 801, 1137, 1347, 1467, 1537, 1713, 1943, 1983, 2103, 2217, 2304, 2305, 2306, 2427, 2643, 2666, 2716, 3867, 3914, 4413

Offset: 1

Neil Fernandez, Dec 21 2004

			143 is in the sequence because it has two unique prime factors (11 and 13), the same number as its two nearest neighbors on both sides (141 has 3 and 47, 142 has 2 and 71, 144 has 2 and 3 and 145 has 5 and 29).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001221, A101932.

For[i=2, i<10000, If[And[Length[FactorInteger[i-2]]==Length[FactorInteger[i]], Length[FactorInteger[i-1]]==Length[FactorInteger[i]], Length[FactorInteger[i+1]]==Length[FactorInteger[i]], Length[FactorInteger[i+2]]==Length[FactorInteger[i]]], Print[i]];i++ ]
Select[Range[600000], PrimeNu[# - 2] == PrimeNu[# - 1] == PrimeNu[#] == PrimeNu[# + 1] == PrimeNu[# + 2] &] (* G. C. Greubel, May 15 2017 *)

Edited by N. J. A. Sloane, Mar 17 2007

A101941 Peak-trough transform of the omega sequence (A001221).

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, -1, 1, -1, 1, -1, 11, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, -1, 11, -1, 0, 0, 0, -1, 1, -1, 0, 0, 0, -3, 0, 0, 0, 0, 0, -1, 5, -1, 0, 0, -2, 0, 3, -2, 0, 0, 3, -1, 1, -1

Offset: 1

Neil Fernandez, Dec 22 2004

sign

I define the peak-trough transform (b(n)) of sequence (c(n)) as follows. If c(n) is greater than both of its immediate neighbors, b(n) is defined as the largest k <= n-1 such that c(n) is greater than its k nearest neighbors both before and after it in the sequence. (c(n) is said to be a k-peak of the sequence). If c(n) is smaller than both of its immediate neighbors, b(n) is defined as -k, where k is the largest k <= n-1 such that c(n) is smaller than its k nearest neighbors both before and after it in the sequence. (c(n) is said to be a k-trough of the sequence). Otherwise, b(n) is 0. (Note difference between a k-peak and a k-apex as defined by Joseph L. Pe in A076759).

Cf. A001221, A023188, A076759, A101932.

Corrected by Peter Munn, May 26 2023

A101936 Numbers n with omega(n) = omega of 3 nearest larger and 3 nearest smaller neighbors.

Original entry on oeis.org

144, 145, 215, 216, 326, 2305, 10283, 16685, 19055, 20215, 21198, 25782, 33335, 35121, 35167, 35205, 39696, 39697, 40272, 41393, 41783, 42345, 42413, 44363, 44364, 44365, 48922, 48923, 48924, 48925, 53737, 54352, 54353, 56017, 56018

Offset: 1

Neil Fernandez, Dec 21 2004

			10283 is in the sequence because 10280, 10281, 10282, 10283, 10284, 10285, 10286 all have the same number of distinct prime factors (3).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001221, A101932, A101932.

For[i=2, i<100000, If[Length[FactorInteger[i-3]] == Length[FactorInteger[i-2]] == Length[FactorInteger[i-1]] == Length[FactorInteger[i]] == Length[FactorInteger[i+1]] == Length[FactorInteger[i+2]] ==Length[FactorInteger[i+3]], Print[i]];i++ ]
Select[Range[1300], PrimeNu[#] == PrimeNu[# - 1] == PrimeNu[# - 2] == PrimeNu[# - 3] == PrimeNu[# + 1] == PrimeNu[# + 2] == PrimeNu[# + 3] &] (* G. C. Greubel, May 21 2017 *)

cp's OEIS Frontend

A076763 1-apexes of omega: numbers n such that omega(n-1) < omega(n) > omega(n+1), where omega(m) = the number of distinct prime factors of m.

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A101934 Numbers n with omega(n) smaller than omega(n-1) and omega(n+1).

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A101294 Numbers n such that omega(n-2) = omega(n-1) = omega(n) = omega(n+1) = omega(n+2).

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A101941 Peak-trough transform of the omega sequence (A001221).

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A101936 Numbers n with omega(n) = omega of 3 nearest larger and 3 nearest smaller neighbors.

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