A045938 -id:A045938 - cp's OEIS frontend

A045932 Numbers n such that n through n+3 are divisible by the same number of distinct primes.

2, 33, 54, 55, 74, 85, 91, 92, 93, 115, 116, 133, 141, 142, 143, 144, 145, 158, 159, 175, 200, 205, 206, 212, 213, 214, 215, 216, 247, 295, 296, 301, 302, 323, 324, 325, 326, 332, 391, 392, 422, 445, 451, 535, 536, 542, 565, 632, 685, 686, 721, 722, 799, 800

Offset: 1

David W. Wilson

nonn

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

Cf. A001221, A006049, A006073, A045933-A045938.

Transpose[Select[Partition[Range[900],4,1],Length[Union[PrimeNu[#]]] == 1&]][[1]] (* Harvey P. Dale, Apr 12 2013 *)

A045933 Numbers n such that n through n+4 are divisible by the same number of distinct primes.

Original entry on oeis.org

54, 91, 92, 115, 141, 142, 143, 144, 158, 205, 212, 213, 214, 215, 295, 301, 323, 324, 325, 391, 535, 685, 721, 799, 1135, 1345, 1465, 1535, 1711, 1941, 1981, 2101, 2215, 2302, 2303, 2304, 2425, 2641, 2664, 2714, 3865, 3912, 4411, 5450, 5461, 6354, 6505

Offset: 1

David W. Wilson

nonn

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

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

SequencePosition[PrimeNu[Range[7000]],{x_,x_,x_,x_,x_}][[All,1]] (* Harvey P. Dale, Jun 13 2022 *)

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

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

A375287 Square array T(n, k), n > 1 and k >= 1, read by antidiagonals in ascending order, 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, 142

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), A002808, A006049 (row 1), A006073, A045932-A045938.

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

cp's OEIS Frontend

A045932 Numbers n such that n through n+3 are divisible by the same number of distinct primes.

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A045933 Numbers n such that n through n+4 are divisible by the same number of distinct primes.

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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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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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A375287 Square array T(n, k), n > 1 and k >= 1, read by antidiagonals in ascending order, 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.

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