A364265 -id:A364265 - 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

A259349 Numbers n such that n-1, n, and n+1 are all products of 6 distinct primes (i.e. belong to A067885).

Original entry on oeis.org

1990586014, 1994837494, 2129658986, 2341714794, 2428906514, 2963553594, 3297066410, 3353808094, 3373085990, 3623442746, 3659230730, 3809238770, 3967387346, 4058711734, 4144727994, 4196154390, 4502893746, 4555267690, 4653623534

Offset: 1

James G. Merickel, Jun 24 2015

nonn

A subsequence of A169834 and A067885.
The rudimentary method employed by the PARI program below reaches the limit of its usefulness here. Contrast it with the method required for A259350, which is over 4.5 orders of magnitude faster than the analog of this (and may still be some distance best).
a(1)=A093550(6) (that sequence's 5th term, with offset 2). The program arbitrarily makes use of this knowledge, but will run (slower) without it.

			1990586013 = 3*13*29*67*109*241,
1990586014 = 2*23*37*43*59*461, and
1990586015 = 5*11*17*19*89*1259; and no smaller trio of this kind exists, making the middle value a(1).

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A093550, A169834, A364265, A248201, A248202, A248203, A248204, A259350, A259801.
For products of 1, 2, 3, 4, 5, and 6 distinct primes see A000040, A006881, A007304, A046386, A046387, and A067885, resp.
See A364265 for a closely related sequence. - N. J. A. Sloane, Jul 18 2023

{
\\Program initialized with known a(1).\\
\\The purpose of vector s and value u\\
\\is to skip bad values modulo 36.\\
k=1990586014;s=[4,4,8,8,8,4];u=1;
while(1,
  if(issquarefree(k),
    if(issquarefree(k-1),
      if(issquarefree(k+1),
        if(omega(k)==6,
          if(omega(k-1)==6,
            if(omega(k+1)==6,
              print1(k" ")))))));
  k+=s[u];if(u==6,u=1,u++))
}

{n: A001221(n-1) = A001221(n) = A001221(n+1) = A001222(n-1) = A001222(n) = A001222(n+1) = 6}. - 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}.

A364308 Numbers k such that k, k+1 and k+2 have exactly 3 distinct prime factors.

Original entry on oeis.org

644, 740, 804, 986, 1034, 1064, 1104, 1220, 1274, 1308, 1309, 1462, 1494, 1580, 1748, 1884, 1885, 1924, 1988, 2013, 2014, 2108, 2134, 2254, 2288, 2294, 2330, 2354, 2364, 2408, 2464, 2484, 2540, 2583, 2584, 2664, 2665, 2666, 2678, 2684, 2714, 2715, 2716, 2754, 2793

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			644 = 2^2*7*23 has 3 distinct prime factors, 645 = 3*5*43 has 3 distinct prime factors, and 646 = 2*17*19 has 3 distinct prime factors, so 644 is in the sequence.

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

Subsequence of A006073 and of A140077.

Cf. A364307 (2 factors), A364309 (4 factors), A364266 (5 factors), A364265 (6 factors), A001221, A080569.

q[n_] := q[n] = PrimeNu[n] == 3; Select[Range[3000], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A080569(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 3}.

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}.

A364436 Numbers that begin a run of at least 4 consecutive integers having exactly 6 distinct prime factors each (i.e., belonging to A074969).

Original entry on oeis.org

7933641735, 9338016258, 9827010633, 10744118592, 10808993635, 10928652579, 13302330390, 15300915705, 16088310249, 16408242849, 18685633314, 18721086153, 19136152098, 19819102092, 20592248544, 20826707802, 21241193334, 21296349633, 21531380583, 21727956885, 21823418253

Offset: 1

David A. Corneth, Jul 24 2023

nonn

			7933641735 is in the sequence as it starts a run of at least 4 consecutive integers each of which has exactly 6 distinct prime factors.
That is, each of 7933641735 = 3 * 5 * 23 * 83 * 461 * 601,
7933641735 + 1 = 7933641736 = 2^3 * 17 * 47 * 59 * 109 * 193,
7933641735 + 2 = 7933641737 = 7 * 29 * 31 * 41 * 97 * 317,
7933641735 + 3 = 2 * 3 * 11 * 89 * 563 * 2399 has 6 distinct prime factors.

Cf. A074969, A364265.

upto(n) = {my(res = List(), streak = 0); n+=3; forfactored(i = 1, n, if(omega(i[2]) == 6, streak++; if(streak >= 4, listput(res, i[1]-3)), streak = 0)); res}

More terms from Jinyuan Wang, Aug 12 2023

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

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A259349 Numbers n such that n-1, n, and n+1 are all products of 6 distinct primes (i.e. belong to A067885).

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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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A364308 Numbers k such that k, k+1 and k+2 have exactly 3 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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A364436 Numbers that begin a run of at least 4 consecutive integers having exactly 6 distinct prime factors each (i.e., belonging to A074969).

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