A364308 -id:A364308 - cp's OEIS frontend

A006073 Numbers k such that k, k+1 and k+2 all have the same number of distinct prime divisors.

2, 3, 7, 20, 33, 34, 38, 44, 50, 54, 55, 56, 74, 75, 85, 86, 91, 92, 93, 94, 98, 115, 116, 117, 122, 133, 134, 141, 142, 143, 144, 145, 146, 158, 159, 160, 175, 176, 183, 187, 200, 201, 205, 206, 207, 212, 213, 214, 215, 216, 217, 224, 235, 247

Offset: 1

N. J. A. Sloane

nonn

Distinct prime divisors means that the prime divisors are counted without multiplicity. - Harvey P. Dale, Apr 19 2011

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A001221, A006049, A045932.

pdQ[n_]:=PrimeNu[n]==PrimeNu[n+1]==PrimeNu[n+2]; Select[Range[250], pdQ] (* Harvey P. Dale, Apr 19 2011 *)
Take[Transpose[Flatten[Select[Partition[{#,PrimeNu[#]}&/@Range[250000], 3,1],#[[1,2]]==#[[2,2]]==#[[3,2]]&],1]][[1]],{1,-1,3}] (* Harvey P. Dale, Dec 09 2011 *)
Flatten[Position[Partition[PrimeNu[Range[250]],3,1],?(#[[1]]==#[[2]]== #[[3]]&),{1},Heads->False]] (* _Harvey P. Dale, Oct 30 2013 *)
SequencePosition[PrimeNu[Range[250]],{x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Jun 12 2024 *)

Union of {2,3,7} and A364307 and A364308 and A364309 and A364266 and A364265 etc. - R. J. Mathar, Jul 18 2023

A364266 The first term in a chain of at least 3 consecutive numbers each with exactly 5 distinct prime factors.

Original entry on oeis.org

1042404, 3460280, 3818828, 3998664, 4638984, 4991964, 5540248, 5701254, 5715500, 5964958, 6772050, 6794084, 7237384, 7453964, 7459088, 7745318, 7757034, 7993194, 8083634, 8153430, 8168194, 8273628, 8340834, 8340980, 8414756, 8486994, 8698898, 8722634, 8758904

Offset: 1

R. J. Mathar, Jul 16 2023

nonn

			1042404 = 2^2*3*11*53*149, 1042405 = 5*6*143*29*79 and 1042406 = 2*17*23*31*43 each have 5 distinct prime factors, so 1042404 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A192203 (subsequence for squarefree triples). Subsequence of A140079 (2 consec.) and of A006073.

Cf. A364308 (3 dist. factors), A364309 (4 dist. factors), A364265 (6 dist. factors), A001221, A087978.

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
for k from 1 do
    if omega(k) = 5 then
        if omega(k+1) = 5 then
            if omega(k+2) = 5 then
                print(k) ;
            end if;
        end if;
    end if;
end do:

seq[lim_] := Module[{s  = {}, q1 = False, q2 = False, q3}, Do[q3 = PrimeNu[k] == 5; If[q1 && q2 && q3, AppendTo[s, k-2]]; q1 = q2; q2 = q3, {k, 3, lim}]; s]; seq[10^7] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A087978(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 5}. - R. J. Mathar, Jul 18 2023

A364307 Numbers k such that k, k+1 and k+2 have exactly 2 distinct prime factors.

Original entry on oeis.org

20, 33, 34, 38, 44, 50, 54, 55, 56, 74, 75, 85, 86, 91, 92, 93, 94, 98, 115, 116, 117, 122, 133, 134, 141, 142, 143, 144, 145, 146, 158, 159, 160, 175, 176, 183, 187, 200, 201, 205, 206, 207, 212, 213, 214, 215, 216, 217, 224, 235, 247, 248, 295, 296

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			44 = 2^2*11 has 2 distinct prime factors, and so has 45 = 3^2*5 and so has 46 = 2*23, so 44 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A006073 and of A074851.
Cf. A364308 (3 factors), A364309 (4 factors), A364266 (5 factors), A364265 (6 factors), A001221.
A039833 is a subsequence.

q[n_] := q[n] = PrimeNu[n] == 2; Select[Range[300], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 2}.

A364309 Numbers k such that k, k+1 and k+2 have exactly 4 distinct prime factors.

Original entry on oeis.org

37960, 44484, 45694, 50140, 51428, 55130, 55384, 61334, 63364, 64294, 67164, 68264, 68474, 70004, 70090, 71708, 72708, 76152, 80444, 81548, 81718, 82040, 84434, 85490, 86240, 90363, 95380, 97382, 98020, 99084, 99384, 99428, 99788, 100164, 100490, 100594, 102254, 102542, 104804, 105994, 108204

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			37960 = 2^3*5*13*73, 37961 = 7*11*17*29, and 37962 = 2*3^3*19*37 each have 4 distinct prime factors, so 37960 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Subsequence of A006073 and of A140078.
A176167 is a subsequence.
Cf. A364307 (2 factors), A364308 (3 factors), A364266 (5 factors), A364265 (6 factors), A001221, A087966, A168628.

q[n_] := q[n] = PrimeNu[n] == 4; Select[Range[10^5], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A087966(3).
a(n)+1 = A168628(n).
{k: A001221(k) = A001221(k+1) = A001221(k+2) = 4}.

cp's OEIS Frontend

A006073 Numbers k such that k, k+1 and k+2 all have the same number of distinct prime divisors.

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A364266 The first term in a chain of at least 3 consecutive numbers each with exactly 5 distinct prime factors.

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A364307 Numbers k such that k, k+1 and k+2 have exactly 2 distinct prime factors.

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A364309 Numbers k such that k, k+1 and k+2 have exactly 4 distinct prime factors.

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