A045920 -id:A045920 - cp's OEIS frontend

A358019 Numbers m such that the factorizations of m..m+10 have the same number of primes (including multiplicities).

202536181, 913535284, 1124342785, 1443929905, 1587749041, 1688485665, 1733574769, 2090053141, 2308638625, 2403102228, 2751673525, 2841766801, 2898584161, 2936217602, 3195380868, 3195380869, 3324630612, 3423884341, 3520752468

Offset: 1

Charles R Greathouse IV, Oct 24 2022

nonn

a(111) = 21117216104 is the first term where the number of primes is 5. - Zak Seidov and Robert Israel, Jun 27 2024

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), this sequence (k=10).

list(lim)=my(v=List(),ct,cur); forfactored(n=202536181,lim\1+10, my(t=bigomega(n)); if(t==cur, if(ct++>9, listput(v,n[1]-10)), cur=t; ct=0)); Vec(v)

A076191 First differences of A001222.

Original entry on oeis.org

1, 0, 1, -1, 1, -1, 2, -1, 0, -1, 2, -2, 1, 0, 2, -3, 2, -2, 2, -1, 0, -1, 3, -2, 0, 1, 0, -2, 2, -2, 4, -3, 0, 0, 2, -3, 1, 0, 2, -3, 2, -2, 2, 0, -1, -1, 4, -3, 1, -1, 1, -2, 3, -2, 2, -2, 0, -1, 3, -3, 1, 1, 3, -4, 1, -2, 2, -1, 1, -2, 4, -4, 1, 1, 0, -1, 1, -2, 4, -1, -2, -1, 3, -2, 0, 0, 2, -3, 3, -2, 1, -1, 0, 0, 4, -5, 2, 0, 1, -3

Offset: 1

Joseph L. Pe, Nov 03 2002

a(A045920(n)) = 0. - Reinhard Zumkeller, Mar 19 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A045920.

a076191 n = a076191_list !! (n-1)
a076191_list = zipWith (-) (tail a001222_list) a001222_list
-- Reinhard Zumkeller, Mar 20 2012

Omega[n_] := Apply[Plus, Transpose[FactorInteger[n]][[2]]]; Flatten[Append[{1}, Table[Omega[n + 1] - Omega[n], {n, 2, 100}]]]

a(n) = bigomega(n + 1) - bigomega(n); \\ Indranil Ghosh, Mar 15 2017

a(n) = Omega(n+1)-Omega(n), where Omega(n) (A001222) denotes the number of prime factors of n, counting multiplicity.

G.f.: ((1 - x)/x)*Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Mar 15 2017

Name changed by Arkadiusz Wesolowski, Jul 27 2012

A045984 a(n) = smallest number m such that factorizations of n consecutive integers starting at m have same number of primes (counted with multiplicity).

Original entry on oeis.org

1, 2, 33, 602, 602, 2522, 211673, 3405122, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465

Offset: 1

David W. Wilson

a(16) > 10^13. a(16) must have at least 5 prime factors (counted with multiplicity) because one of the 16 consecutive numbers is divisible by 2^4. - Donovan Johnson, Apr 01 2013

			a(4) = 602 as 602 = 2 * 7 * 43, 603 = 3 * 3 * 67, 604 = 2 * 2 * 151, 605 = 5 * 11 * 11 so four consecutive positive integers have the same number of prime factors starting at 602, the first such number. - _David A. Corneth_, Feb 24 2024

Cf. A001222, A045920, A045939, A045940, A045941, A045942, A077657, A117969, A356953.

More terms from Vladeta Jovovic, Aug 06 2002

More terms from Martin Fuller, Nov 21 2006

A113752 Smallest number m such that m, m+1 and m+2 have exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

33, 170, 1274, 15470, 33614, 3145310, 40909374, 668363967, 9864741248, 179199427328, 967461818750, 57938945781248, 597779386906624

Offset: 2

Martin Fuller, Jan 17 2006

n = A001222(a(n)) = A001222(a(n)+1) = A001222(a(n)+2). Subsequence of A045920.
a(14) <= 1247579465781248. - Donovan Johnson, Jun 12 2013
a(15) > 2 * 10^15. - Toshitaka Suzuki, Aug 31 2025

			a(6) = 33614 = 2*7*7*7*7*7, a(6)+1 = 3*3*3*3*5*83, a(6)+2 = 2*2*2*2*11*191

Cf. A001222, A045920, A093549, A093550, A115186.

t = {}; n = 2; m = 1; While[Length[t] < 5, m++; If[n == PrimeOmega[m] == PrimeOmega[m + 1] == PrimeOmega[m + 2], AppendTo[t, m]; n++]]; t (* T. D. Noe, Aug 19 2013 *)

a(9)-a(11) from Donovan Johnson, Apr 08 2008
a(12) from Donovan Johnson, Aug 08 2011
a(13) from Jud McCranie, Aug 19 2013
a(14) from Toshitaka Suzuki, Aug 31 2025

A077657 Least number with exactly n consecutive successors, all having the same number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 33, 603, 602, 2522, 211673, 3405123, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465

Offset: 0

Reinhard Zumkeller, Nov 13 2002

A001222(a(n))=A001222(a(n)+k) for k<=n;

A077655(a(n))=n and A077655(k)

			a(0)=A077656(1)=1; a(1)=A045920(1)=2; a(2)=A045939(1)=33; a(3)=A045940(2)=603; a(4)=A045941(1)=602; a(5)=A045942(1)=2522.

Cf. A045984.

a(n)=A045984(n+1)+A077655(A045984(n+1))-n - Martin Fuller, Nov 21 2006

More terms from Martin Fuller, Nov 21 2006

A280382 Numbers k such that k-1 has the same number of prime factors counted with multiplicity as k+1.

Original entry on oeis.org

4, 5, 6, 12, 18, 19, 29, 30, 34, 42, 43, 50, 51, 55, 56, 60, 67, 69, 72, 77, 86, 89, 92, 94, 102, 108, 115, 120, 122, 138, 142, 144, 150, 151, 160, 171, 173, 180, 184, 186, 187, 189, 192, 197, 198, 202, 204, 214, 216, 218, 220, 228, 233, 236, 237, 240, 243, 245, 248, 249, 266, 267, 270, 271, 274, 282

Offset: 1

Rick L. Shepherd, Jan 01 2017

nonn

			Unlike for A088070, 5 is a term here because 4 = 2^2 and 6 = 2*3 each have two prime factors when counted with multiplicity. Similarly, 3 is not a term of this sequence (but is in A088070) because 2 and 4 have different numbers of prime factors as counted by A001222.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A001222, A088070 (similar but prime factors counted without multiplicity), A280383 (prime factor count is same both ways), A280469 (subsequence of current with k-1 and k+1 squarefree also), A045920 (similar but for k and k+1).

Cf. A115167 (subsequence of odd terms).

Select[Range[2, 300], Equal @@ PrimeOmega[# + {-1, 1}] &] (* Amiram Eldar, May 20 2021 *)

IsInA280382(n) = n > 1 && bigomega(n-1) == bigomega(n+1)

from sympy import primeomega
def aupto(limit):
  prv, cur, nxt, alst = 1, 1, 2, []
  for n in range(3, limit+1):
    if prv == nxt: alst.append(n)
    prv, cur, nxt = cur, nxt, primeomega(n+2)
  return alst
print(aupto(282)) # Michael S. Branicky, May 20 2021

A322838 Number of positive integers less than n with more prime factors than n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 5, 0, 6, 2, 2, 0, 9, 1, 10, 1, 5, 5, 13, 0, 6, 6, 2, 2, 18, 2, 19, 0, 10, 10, 10, 1, 24, 11, 11, 1, 27, 5, 28, 5, 5, 15, 31, 0, 16, 6, 17, 6, 36, 2, 19, 2, 20, 20, 41, 2, 42, 21, 9, 0, 23, 10, 47, 10, 25, 10, 50, 1, 51, 27, 11, 11

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

			Column n lists the a(n) positive integers less than n with more prime factors than n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
              4     6     8  8   10      12  12  12      16  16  18  16
                    4            9       10  8   8       15      16
                                 8       9               14      15
                                 6       8               12      14
                                 4       6               10      12
                                         4               9       10
                                                         8       9
                                                         6       8
                                                         4       6
                                                                 4

Positions of zeros appear to be A029744.

Cf. A000010, A000961, A001221, A001222, A045920, A058933, A061142, A067003, A302242, A322839, A322841.

Table[Length[Select[Range[n],PrimeOmega[#]>PrimeOmega[n]&]],{n,100}]

A278291 Numbers n such that n-1 has the same number of prime factors as n (with multiplicity).

Original entry on oeis.org

3, 10, 15, 22, 26, 28, 34, 35, 39, 45, 58, 76, 86, 87, 94, 95, 99, 117, 119, 122, 123, 125, 134, 136, 142, 143, 146, 148, 154, 159, 165, 171, 172, 175, 178, 202, 203, 206, 214, 215, 218, 219, 231, 245, 246, 254, 285, 286, 297, 299, 302, 303, 327, 333, 335, 351, 357, 362, 370, 376, 382, 388, 394, 395

Offset: 1

Ely Golden, Nov 16 2016

nonn

			a(1)=3, as both 2 and 3 have 1 prime factor. a(2)=10, as both 9 and 10 have 2 prime factors. a(3)=15, as both 14 and 15 have 2 prime factors.

Ely Golden, Table of n, a(n) for n = 1..10000

public class A278291{
public static void main(String[] args)throws Exception{
    long dim0=numberOfPrimeFactors(2);//note that this method must be manually implemented by the user
    long dim1;
    long counter=3;
    long index=1;
    while(index<=10000){
      dim1=numberOfPrimeFactors(counter);
      if(dim1==dim0){System.out.println(index+" "+counter);index++;}
      dim0=dim1;
      counter++;
    }
  }
}

fQ[n_] := PrimeOmega[n - 1] == PrimeOmega[n]; Select[Range@400, fQ] (* Robert G. Wilson v, Nov 17 2016 *)

is(n) = bigomega(n)==bigomega(n-1) \\ Felix Fröhlich, Nov 17 2016

def bigomega(x):
    s=0;
    f=list(factor(x));
    for c in range(len(f)):
        s+=f[c][1]
    return s;
dim0=bigomega(2);
counter=3
index=1
while(index<=10000):
    dim1=bigomega(counter);
    if(dim1==dim0):
        print(str(index)+" "+str(counter))
        index+=1;
    dim0=dim1;
    counter+=1;

a(n) = A045920(n) + 1. - Robert G. Wilson v, Nov 17 2016

A322839 Numbers n with more prime factors (counted with multiplicity) than n+1.

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 45, 46, 48, 50, 52, 54, 56, 58, 60, 64, 66, 68, 70, 72, 76, 78, 80, 81, 82, 84, 88, 90, 92, 96, 100, 102, 104, 105, 106, 108, 110, 112, 114, 117, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

First differs from A074827 in having 104.

			104 has four prime factors (2, 2, 2, 13), while 105 has only three (3, 5, 7), so 104 belongs to the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000961, A001221, A001222, A045920, A294277, A294278, A302242, A322838, A322840.

R:= NULL: count:= 0: w:= 0:
for n from 2 while count < 100 do
  v:= numtheory:-bigomega(n);
  if v < w then R:= R,n-1; count:= count+1; fi;
  w:= v;
od:
R; # Robert Israel, Jan 16 2023

Select[Range[100],PrimeOmega[#]>PrimeOmega[#+1]&]
Position[Partition[PrimeOmega[Range[200]],2,1],?(#[[1]]>#[[2]]&),1,Heads -> False]//Flatten (* _Harvey P. Dale, Jan 02 2021 *)

A358817 Numbers k such that A046660(k) = A046660(k+1).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, Dec 02 2022

nonn

First differs from its subsequence A007674 at n=18.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 38, 369, 3655, 36477, 364482, 3644923, 36449447, 364494215, 3644931537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.36449... .

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

Cf. A046660.
Subsequences: A007674, A052213, A085651, A358818.
Similar sequences: A002961, A005237, A006049, A045920.

seq[kmax_] := Module[{s = {}, e1 = 0, e2}, Do[e2 = PrimeOmega[k] - PrimeNu[k]; If[e1 == e2, AppendTo[s, k - 1]]; e1 = e2, {k, 2, kmax}]; s]; seq[160]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
lista(nmax) = {my(e1 = e(1), e2); for(n=2, nmax, e2=e(n); if(e1 == e2, print1(n-1,", ")); e1 = e2);}

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cp's OEIS Frontend

A358019 Numbers m such that the factorizations of m..m+10 have the same number of primes (including multiplicities).

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A076191 First differences of A001222.

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A045984 a(n) = smallest number m such that factorizations of n consecutive integers starting at m have same number of primes (counted with multiplicity).

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A113752 Smallest number m such that m, m+1 and m+2 have exactly n prime factors (counted with multiplicity).

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A077657 Least number with exactly n consecutive successors, all having the same number of prime factors (counted with multiplicity).

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A280382 Numbers k such that k-1 has the same number of prime factors counted with multiplicity as k+1.

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A322838 Number of positive integers less than n with more prime factors than n, counted with multiplicity.

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A278291 Numbers n such that n-1 has the same number of prime factors as n (with multiplicity).

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