A045932 -id:A045932 - 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

A045940 Numbers m such that the factorizations of m..m+3 have the same number of primes (including multiplicities).

Original entry on oeis.org

602, 603, 1083, 2012, 2091, 2522, 2523, 2524, 2634, 2763, 3243, 3355, 4023, 4202, 4203, 4921, 4922, 4923, 5034, 5035, 5132, 5203, 5282, 5283, 5785, 5882, 5954, 5972, 6092, 6212, 6476, 6962, 6985, 7314, 7730, 7731, 7945, 8393, 8825, 8956, 8972, 9162

Offset: 1

David W. Wilson

nonn

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..11998 (First 3356 terms from Zak Seidov)

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), this sequence (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).

Cf. A045932 (similar, with omega).

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};lst={};Do[If[f[n]==f[n+1]==f[n+2]==f[n+3],AppendTo[lst,n]],{n,0,8!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
SequencePosition[PrimeOmega[Range[10000]],{x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 02 2020 *)

isok(n) = (bigomega(n) == bigomega(n+1)) && (bigomega(n+1) == bigomega(n+2)) && (bigomega(n+2) == bigomega(n+3)); \\ Michel Marcus, Jan 06 2015

A045938 Numbers n such that n through n+9 are divisible by the same number of distinct primes.

Original entry on oeis.org

48919, 184171, 218972, 218973, 320085, 320671, 343443, 353944, 397322, 403117, 435721, 492037, 526095, 526096, 526097, 526098, 526099, 534078, 534079, 534080, 583340, 607116, 636332, 693841, 701595, 761492, 822260, 919998, 942528

Offset: 1

David W. Wilson

nonn

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

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

Offset corrected by Amiram Eldar, Oct 26 2019

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

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

A338454 Starts of runs of 4 consecutive numbers with the same total binary weight of their divisors (A093653).

Original entry on oeis.org

242, 947767, 1041607, 2545015, 3275463, 8170983, 15720871, 21532430, 23752181, 25135885, 25595913, 27981703, 28226983, 30505142, 30962767, 33364805, 37264493, 49002661, 49766629, 52910454, 53408456, 57917191, 57952016, 58331576, 59230454, 60014053, 60723111, 63378005

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1) = A093653(k+2) = A093653(k+3).

			242 is a term since A093653(242) = A093653(243) = A093653(244) = A093653(245) = 18.

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

Cf. A093653.
Subsequence of A338452 and A338453.
Similar sequences: A006601, A045932, A045940.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; m = 4; fs = f /@ Range[m]; Do[If[Equal @@  fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 10^7}]; s

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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A045940 Numbers m such that the factorizations of m..m+3 have the same number of primes (including multiplicities).

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A045938 Numbers n such that n through n+9 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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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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