A006073 -id:A006073 - cp's OEIS frontend

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

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}.

A045934 Numbers n such that n through n+5 have the same number of distinct prime factors.

Original entry on oeis.org

91, 141, 142, 143, 212, 213, 214, 323, 324, 2302, 2303, 6850, 9061, 10280, 10281, 15740, 16130, 16164, 16682, 16683, 19052, 19053, 20212, 20213, 21195, 21196, 21790, 22055, 23064, 25779, 25780, 25991, 28608, 28674, 29971, 31442, 33084

Offset: 1

David W. Wilson

nonn

			The numbers from 91 to 96 all have 2 distinct prime factors: 91=7*13, 92=2^2*23, 93=3*31, 94=2*47, 95=5*19, and 96=2^5*3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2620 from Zak Seidov)
Roger B. Eggleton and James A. MacDougall, Consecutive integers with equally many principal divisors, Math. Mag. 81 (2008), 235-248.

Cf. A001221, A006049, A006073, A045932-A045933, A045935-A045938, A088983.

Select[Range[35000],Length[Union[Length/@FactorInteger[Range[#,#+5]]]]==1&]  (* Harvey P. Dale, Feb 27 2011 *)

A045935 Numbers n such that n through n+6 are divisible by the same number of distinct primes.

Original entry on oeis.org

141, 142, 212, 213, 323, 2302, 10280, 16682, 19052, 20212, 21195, 25779, 33332, 35118, 35164, 35202, 39693, 39694, 40269, 41390, 41780, 42342, 42410, 44360, 44361, 44362, 48919, 48920, 48921, 48922, 53734, 54349, 54350, 56014, 56015

Offset: 1

David W. Wilson

nonn

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

Cf. A001221, A006049, A006073, A045932-A045934, A045936-A045938.

Offset corrected by Amiram Eldar, Oct 26 2019

A075046 a(n) = the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are in nondescending order.

Original entry on oeis.org

1, 1, 1, 1, 241, 241, 12853, 12853, 234613, 376741, 78312721, 125938261, 4019167441, 16586155153, 35237422882, 1296230533473, 42301168491121, 61118966262061

Offset: 1

Amarnath Murthy, Sep 03 2002

tau(k) <= tau(k+1) <= ... <= tau(k+n-1).
a(16) > 10^12. - Donovan Johnson, Oct 13 2009
a(17) > 10^13. - Giovanni Resta, Apr 12 2017
a(19) > 2.64*10^15. - Jud McCranie, Mar 27 2019
If a(n) > 1, then A013632(a(n)) >= n. Might be useful to help speed up brute force search. - Chai Wah Wu, May 04 2017

			a(5) = 241 = a(6) as tau(241) = 2 < tau(242) = tau(243) = tau(244) = tau(245) = 6 < tau(246).

Cf. A075028, A075029, A075031, A045983, A006073, A075044, A055927, A075047.

k = 1; Do[ While[t = Table[ DivisorSigma[0, i], {i, k, k + n - 1}]; t != Sort[t], k++ ]; Print[k], {n, 1, 11}]

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
a(11) from Robert G. Wilson v, Sep 07 2003
a(12)-a(15) from Donovan Johnson, Oct 13 2009
a(16) from Fred Schneider, Mar 29 2017
a(17)-a(18) from Jud McCranie, Mar 27 2019

A045936 Numbers n such that n through n+7 are divisible by the same number of distinct primes.

Original entry on oeis.org

141, 212, 39693, 44360, 44361, 48919, 48920, 48921, 54349, 56014, 56015, 56791, 60044, 65721, 72650, 72651, 73292, 73293, 76581, 76582, 82324, 82325, 86331, 86332, 87758, 87759, 90092, 91814, 91815, 99843, 106249, 112142, 112143, 121594

Offset: 1

David W. Wilson

nonn

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

Cf. A001221, A006049, A006073, A045932-A045935, A045937-A045938.

npQ[n_]:=Length[Union[Length[FactorInteger[#]]&/@Range[n,n+7]]]==1
Select[Range[125000],npQ]  (* Harvey P. Dale, Feb 23 2011 *)

Offset corrected by Amiram Eldar, Oct 26 2019

A045937 Numbers n such that n through n+8 are divisible by the same number of distinct primes.

Original entry on oeis.org

44360, 48919, 48920, 56014, 72650, 73292, 76581, 82324, 86331, 87758, 91814, 112142, 143491, 147951, 158719, 184171, 184172, 197588, 202498, 205244, 215300, 218972, 218973, 218974, 229728, 230628, 241129, 250933, 253204, 253665, 287492

Offset: 1

David W. Wilson

nonn

Primes counted without multiplicity. - Harvey P. Dale, May 05 2015

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

Cf. A001221, A006049, A006073, A045932-A045936, A045938.

Flatten[Position[Partition[PrimeNu[Range[300000]],9,1],?(Length[ Union[ #]] == 1&),{1},Heads->False]] (* _Harvey P. Dale, May 05 2015 *)

Offset corrected by Amiram Eldar, Oct 26 2019

A088983 Numbers n such that each of the 6 consecutive numbers n through n+5 has exactly two distinct prime factors.

Original entry on oeis.org

91, 141, 142, 143, 212, 213, 214, 323, 324, 2302, 2303

Offset: 1

Labos Elemer, Sep 30 2003

Initial segment of A045934 is identical to this sequence but in A045934 the 12th term is divisible by 3 prime factors. Is the present sequence complete?
No more terms < 3*10^8. - David Wasserman, Aug 29 2005
a(12) > 10^40, if it exists. - Giovanni Resta, May 10 2017
From David A. Corneth, May 14 2017: (Start)
We're looking for at least 6 consecutive positive integers that each have exactly two distinct prime divisors. I.e. 6 consecutive positive integers m with omega(m) = 2. Now of exactly 6 consecutive integers, exactly one of them is divisible by 6, i.e. m is of the form 2*3*k. However m has exactly 2 distinct prime divisors, so k can only have prime divisors 2 or 3. Now, suppose m ends in 6 or higher. Then one of the consecutive integers is divisible by 10 = 2*5. I.e. it's of the form 2*5*t. Then t can only have prime divisors 2 and 5. (End)
This sequence has no run of four consecutive integers, since Eggleton and MacDougall prove that there are no more than 9 consecutive integers with A001221(k) = 2. They conjecture that A007774 contains no runs of 9 consecutive integers, and has only two runs of size 8 (at 141 and 212) and two maximal runs of size 7 (at 323 and 2302); they add that the maximal run of size 6 at 91 might be the only such run, so A088983 might be complete. - Roger Eggleton via Jason Kimberley, Jul 12 2017

Roger B. Eggleton and James A. MacDougall, Consecutive integers with equally many principal divisors, Math. Mag. 81 (2008), 235-248.

Cf. A064709, A045932, A045933, A045934, A045935, A074851, A006073, A003586.

Select[Range[3000], AllTrue[# + Range[0, 5], Length@FactorInteger[#] == 2 &] &] (* Giovanni Resta, May 09 2017 *)

Definition simplified by Roger Eggleton via Jason Kimberley, Jul 12 2017

A355711 Starts of runs of 3 consecutive numbers with the same number of 5-smooth divisors.

Original entry on oeis.org

33, 85, 93, 145, 213, 265, 393, 445, 453, 475, 505, 633, 685, 753, 805, 813, 865, 933, 985, 993, 1045, 1113, 1165, 1293, 1345, 1353, 1405, 1430, 1533, 1585, 1624, 1653, 1705, 1713, 1765, 1833, 1885, 1893, 1945, 2013, 2065, 2193, 2245, 2253, 2275, 2305, 2433, 2485

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355583(k) = A355583(k+1) = A355583(k+2).

			33 is a term since A355583(33) = A355583(34) = A355583(35) = 2.

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

Cf. A355583.
Subsequence of A355710.
A355712 is a subsequence.
Similar sequences: A005238, A006073, A045939, A332313, A332387.

f[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); s = {}; m = 3; fs = f /@ Range[m]; Do[If[Equal @@ fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 2500}]; s

s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
s1 = s(1); s2 = s(2); for(k = 3, 2500, s3 = s(k); if(s1 == s2 && s2 == s3, print1(k-2,", ")); s1 = s2; s2 = s3);

A075044 a(0) = 1; a(n) = the smallest number k such that n numbers from k to k+n-1 have n distinct prime divisors, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 14, 644, 134043, 129963314, 626804494291

Offset: 0

Amarnath Murthy, Sep 03 2002

a(7) <= 45164156742722455667280. - Giorgos Kalogeropoulos, Apr 10 2025

Cf. A006049, A045983, A006073.

Do[k = 1; While[Length /@ FactorInteger /@ Range[k, k+n-1] != Table[n, {n}], k++ ]; Print[k], {n, 0, 5}] (* Ryan Propper, Oct 01 2005 *)

Corrected and extended by Ryan Propper, Oct 01 2005

Offset corrected and a(6) from Donovan Johnson, Aug 03 2009

A375287 Square array T(n, k), n > 1 and k >= 1, read by upward antidiagonals, give the smallest number that starts a sequence of exactly k consecutive numbers, each having exactly n distinct prime factors (counted without multiplicity), or -1 if no such number exists.

Original entry on oeis.org

6, 30, 14, 210, 230, 20, 2310, 7314, 644, 33, 30030, 254540, 37960, 1308, 54, 510510, 11243154, 1042404, 134043, 2664, 91, 9699690, 965009045, 323567034, 21871365, 357642, 6850, 323, 223092870, 65893166030, 30989984674, 7933641735, 129963314, 2713332, 10280, 141

Offset: 2

Jean-Marc Rebert, Aug 10 2024

All positive terms are composite.

			T(2,3) = 20 = 2^2 * 5, because both 21 and 22 have the same omega. Thus, 20 is the starting number of a run of 3 numbers that each have same omega, i.e. 2. No lesser number has this property, so T(2,3) = 20.
Table begins (upper left corner = T(2,1)):
      6       14        20         33 ...
     30      230       644       1308 ...
    210     7314     37960     134043 ...
   2310   254540   1042404   21871365 ...
  30030 11243154 323567034 7933641735 ...
    ...      ...       ...        ...

Cf. A001221, A002110 (col 1), A006049, A006073, A045932-A045938, A064709 (row 2), A185032 (row 3), A185042 (row 4), A384507 (row 5).

T(n,1) = A002110(n) for n > 1.

a(29) corrected by and a(30)-a(37) from Jinyuan Wang, Sep 05 2025

cp's OEIS Frontend

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

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A045934 Numbers n such that n through n+5 have the same number of distinct prime factors.

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A045935 Numbers n such that n through n+6 are divisible by the same number of distinct primes.

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A075046 a(n) = the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are in nondescending order.

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A045936 Numbers n such that n through n+7 are divisible by the same number of distinct primes.

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A045937 Numbers n such that n through n+8 are divisible by the same number of distinct primes.

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A088983 Numbers n such that each of the 6 consecutive numbers n through n+5 has exactly two distinct prime factors.

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A355711 Starts of runs of 3 consecutive numbers with the same number of 5-smooth divisors.

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A075044 a(0) = 1; a(n) = the smallest number k such that n numbers from k to k+n-1 have n distinct prime divisors, or 0 if no such number exists.

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