A006049 -id:A006049 - cp's OEIS frontend

A107800 a(n) = number of distinct primes dividing A006049(n) (and dividing A006049(n)+1).

1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2

Offset: 1

Leroy Quet, Mar 24 2007

nonn

a(n) first equals 3 when n is such that A006049(n) = 230.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from T. D. Noe)

Cf. A001221, A006049, A052215.

f[n_] := Length@FactorInteger[n];t = f /@ Range[300];f /@ Flatten@Position[Rest[t] - Most[t], 0] (* Ray Chandler, Mar 27 2007 *)

a(n) = A001221(A006049(n)).

Extended by Ray Chandler, Mar 27 2007

A075041 Erroneous version of A006049.

Original entry on oeis.org

2, 14, 16, 20, 21, 31, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87

Offset: 1

Amarnath Murthy, Sep 03 2002

dead

This appears to be either an erroneous version of A045920 or an erroneous version of A006049. - R. J. Mathar, Aug 13 2012

Cf. A045983.

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

Original entry on oeis.org

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

A045983 Numbers k such that n or more consecutive integers starting at k have the same number of distinct prime divisors.

Original entry on oeis.org

1, 2, 2, 2, 54, 91, 141, 141, 44360, 48919, 218972, 526095, 526095, 526095, 17233173, 127890362, 29138958036, 118968284928, 118968284928, 585927201062, 585927201062, 585927201062, 585927201062, 313978488186061, 453918847597184, 453918847597184, 455626105596320

Offset: 1

David W. Wilson

nonn

a(n) = smallest number k such that the n numbers from k through n+k-1 have the same number of prime divisors.
a(24) > 10^12. - Donovan Johnson, Mar 29 2013
a(28) > 2 * 10^15. - Toshitaka Suzuki, Jun 22 2025

			a(5) = 54 as 54, 55, 56, 57, 58 all have 2 prime divisors.

Cf. A006049, A075028, A075031.

v=vector(16); n=0; c1=0; for(k=1, 127890377, c2=omega(k); if(c1==c2, n++; if(v[n]==0, v[n]=k-n+1; print(n " " v[n])), n=1; c1=c2)) /* Donovan Johnson, Mar 29 2013 */

a(18)-a(23) from Donovan Johnson, Mar 29 2013

a(24)-a(27) from Toshitaka Suzuki, Jun 22 2025

A058933 Let k be bigomega(n) (i.e., n is a k-almost-prime). a(n) = number of k-almost-primes <= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 3, 4, 5, 2, 6, 5, 6, 1, 7, 3, 8, 4, 7, 8, 9, 2, 9, 10, 5, 6, 10, 7, 11, 1, 11, 12, 13, 3, 12, 14, 15, 4, 13, 8, 14, 9, 10, 16, 15, 2, 17, 11, 18, 12, 16, 5, 19, 6, 20, 21, 17, 7, 18, 22, 13, 1, 23, 14, 19, 15, 24, 16, 20, 3, 21, 25, 17, 18, 26, 19, 22, 4, 8, 27, 23

Offset: 1

Naohiro Nomoto, Jan 11 2001

Equivalently, the number of positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity. - Gus Wiseman, Dec 28 2018

There is a close relationship between a(n) and a(n^2). See A209934 for an exploratory quantification. - Peter Munn, Aug 04 2019

			3 is prime, so a(3)=2. 10 is 2-almost prime (semiprime), so a(10)=4.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
        2     3  4  5     6  9   7   8   11  10  14      13  12  17  18
              2     3     4  6   5       7   9   10      11  8   13  12
                    2        4   3       5   6   9       7       11  8
                                 2       3   4   6       5       7
                                         2       4       3       5
                                                         2       3
                                                                 2
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Positions of 1's are A000079.
Cf. A001358, A014612, A014613, A014614.
Cf. A000010, A000961, A001222, A006049, A045920, A061142, A067003, A078843, A209934, A302242, A322838, A322839, A322840.
Equivalent sequence restricted to squarefree numbers: A340313.

p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= numtheory[bigomega](n);
      p(t):= p(t)+1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2015

p[] = 0; a[n] := a[n] = Module[{t}, t = PrimeOmega[n]; p[t] = p[t]+1]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 24 2017, after Alois P. Heinz *)

a(n) = my(k=bigomega(n)); sum(i=1, n, bigomega(i)==k); \\ Michel Marcus, Jun 27 2024

from math import prod, isqrt
from sympy import isprime, primepi, primerange, integer_nthroot, primeomega
def A058933(n):
    if n==1: return 1
    if isprime(n): return primepi(n)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,primeomega(n)))) # Chai Wah Wu, Aug 28 2024

Ordinal transform of A001222 (bigomega). - Franklin T. Adams-Watters, Aug 28 2006

If a(n) < a(3^A001222(2n)) = A078843(A001222(2n)) then a(2n) = a(n), otherwise a(2n) > a(n). - Peter Munn, Aug 05 2019

Name edited by Peter Munn, Dec 30 2022

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

Original entry on oeis.org

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 *)

A074851 Numbers k such that k and k+1 both have exactly 2 distinct prime factors.

Original entry on oeis.org

14, 20, 21, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 111, 115, 116, 117, 118, 122, 123, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 152, 158, 159, 160, 161, 171, 175, 176, 177, 183, 184

Offset: 1

Benoit Cloitre, Sep 10 2002

Subsequence of A006049. - Michel Marcus, May 06 2016

			20=2^2*5 21=3*7 hence 20 is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A006049, A006549, A001221.
Analogous sequences for m distinct prime factors: this sequence (m=2), A140077 (m=3), A140078 (m=4), A140079 (m=5), A273879 (m=6).
Cf. A093548.
Equals A255346 \ A321502.

Filtered([1..200],n->[Size(Set(Factors(n))),Size(Set(Factors(n+1)))]=[2,2]); # Muniru A Asiru, Dec 05 2018

[n: n in [2..200] | #PrimeDivisors(n) eq 2 and #PrimeDivisors(n+1) eq 2]; // Vincenzo Librandi, Dec 05 2018

Flatten[Position[Partition[Table[If[PrimeNu[n]==2,1,0],{n,200}],2,1],{1,1}]] (* Harvey P. Dale, Mar 12 2015 *)

isok(n) = (omega(n) == 2) && (omega(n+1) == 2); \\ Michel Marcus, May 06 2016

import sympy
from sympy.ntheory.factor_ import primenu
for n in range(1,200):
    if primenu(n)==2 and primenu(n+1)==2:
        print(n, end=', '); # Stefano Spezia, Dec 05 2018

a(n) seems to be asymptotic to c*n*log(n)^2 with c=0.13...

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

A067003 Number of numbers <= n with same number of distinct prime factors as n.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 5, 6, 7, 2, 8, 3, 9, 4, 5, 10, 11, 6, 12, 7, 8, 9, 13, 10, 14, 11, 15, 12, 16, 1, 17, 18, 13, 14, 15, 16, 19, 17, 18, 19, 20, 2, 21, 20, 21, 22, 22, 23, 23, 24, 25, 26, 24, 27, 28, 29, 30, 31, 25, 3, 26, 32, 33, 27, 34, 4, 28, 35, 36, 5, 29, 37, 30, 38, 39, 40, 41

Offset: 1

Henry Bottomley, Dec 21 2001

			a(11)=8 since 2,3,4,5,7,8,9,11 each have one distinct prime factor. a(12)=3 since 6,10,12 each have two distinct prime factors.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of distinct prime factors as n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
        2  3  4     5  7  8  6   9   10  11  12  14  13  16  15  17  18
           2  3     4  5  7      8   6   9   10  12  11  13  14  16  15
              2     3  4  5      7       8   6   10  9   11  12  13  14
                    2  3  4      5       7       6   8   9   10  11  12
                       2  3      4       5           7   8   6   9   10
                          2      3       4           5   7       8   6
                                 2       3           4   5       7
                                         2           3   4       5
                                                     2   3       4
                                                         2       3
                                                                 2
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Positions of 1's are A002110.
Cf. A001221, A008479, A058933, A067004. Inverse of A000961, A007774, A033992, A033993, A051270 etc.
Cf. A000010, A006049, A061142, A294277, A294278, A302242, A322837, A322841.

Table[Length[Select[Range[n],PrimeNu[#]==PrimeNu[n]&]],{n,100}] (* Gus Wiseman, Dec 28 2018 *)

a(n) = my(nb = #factor(n)~); sum(k=1, n, #factor(k)~ == nb); \\ Michel Marcus, Jul 13 2019

a(A002110(n)) = 1.

A343819 Numbers k such that k and k+1 have the same number of Fermi-Dirac factors (A064547).

Original entry on oeis.org

2, 3, 4, 14, 16, 20, 21, 26, 27, 32, 33, 34, 35, 38, 44, 45, 50, 51, 57, 62, 63, 64, 68, 74, 75, 76, 85, 86, 91, 92, 93, 94, 98, 99, 104, 111, 115, 116, 117, 118, 122, 123, 124, 133, 135, 141, 142, 143, 144, 145, 146, 147, 158, 161, 171, 175, 176, 177, 187, 189

Offset: 1

Amiram Eldar, Apr 30 2021

nonn

Since the number of infinitary divisors of k is A037445(k) = 2^A064547(k), this is also the sequence of numbers k such that k and k+1 have the same number of infinitary divisors.

			2 is a term since A064547(2) = A064547(3) = 1.

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

Cf. A064547, A306985, A343818.
Similar sequences: A005237, A006049.
Subsequence of A086263.

fd[1] = 0; fd[n_] := Plus @@ DigitCount[FactorInteger[n][[;;,2]], 2, 1]; Select[Range[200], fd[#] == fd[# + 1] &]

A294277 Numbers k such that omega(k) < omega(k+1) (where omega(m) = A001221(m), the number of distinct primes dividing m).

Original entry on oeis.org

1, 5, 9, 11, 13, 17, 19, 23, 25, 27, 29, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 89, 97, 101, 103, 104, 107, 109, 113, 119, 121, 125, 128, 129, 131, 137, 139, 149, 151, 153, 155, 157, 163, 164, 167, 169, 173, 179, 181, 185

Offset: 1

Rémy Sigrist, Oct 26 2017

This sequence, alongside A006049 and A294278, form a partition of the positive integers.

The asymptotic density of this sequence is 1/2 (Erdős, 1936). - Amiram Eldar, Sep 17 2024

			omega(1) = 0 < omega(2) = 1, hence 1 belongs to this sequence.
omega(4) = 1 = omega(5) = 1, hence 4 does not belong to this sequence.
omega(6) = 2 > omega(7) = 1, hence 6 does not belong to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Cf. A001221, A006049, A294278.

Position[Partition[PrimeNu[Range[200]],2,1],?(#[[1]]<#[[2]]&),1,Heads-> False]//Flatten (* _Harvey P. Dale, May 06 2018 *)

for (n=1, 185, if (omega(n) < omega(n+1), print1 (n ", ")))

cp's OEIS Frontend

A107800 a(n) = number of distinct primes dividing A006049(n) (and dividing A006049(n)+1).

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A075041 Erroneous version of A006049.

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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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A045983 Numbers k such that n or more consecutive integers starting at k have the same number of distinct prime divisors.

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A058933 Let k be bigomega(n) (i.e., n is a k-almost-prime). a(n) = number of k-almost-primes <= n.

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

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A074851 Numbers k such that k and k+1 both have exactly 2 distinct prime factors.

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A067003 Number of numbers <= n with same number of distinct prime factors as n.

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