A115186 -id:A115186 - cp's OEIS frontend

A045920 Numbers m such that the factorizations of m..m+1 have the same number of primes (including multiplicities).

2, 9, 14, 21, 25, 27, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 116, 118, 121, 122, 124, 133, 135, 141, 142, 145, 147, 153, 158, 164, 170, 171, 174, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 245, 253, 284, 285, 296, 298, 301, 302, 326, 332, 334, 350, 356, 361

Offset: 1

Felice Russo

A115186 is a subsequence: A001222(A115186(n)) = A001222(A115186(n)+1) = n. - Reinhard Zumkeller, Jan 16 2006
Indices k such that A076191(k) = 0. - Ray Chandler, Dec 10 2008
A045939 is a subsequence. - Zak Seidov, Jul 02 2020
This sequence is infinite (Heath-Brown, 1984). - Amiram Eldar, Jul 11 2020

C. Clawson, Mathematical mysteries, Plenum Press 1996, p. 250.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, A parity problem from sieve theory, Mathematika, Vol. 29, No. 1 (1982), pp. 1-6.
D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika, Vol. 31, No. 1 (1984), pp. 141-149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific journal of mathematics, Vol. 129, No. 2 (1987), pp. 307-319.

Numbers m through m+k have the same number of prime divisors (with multiplicity): this sequence (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), A358019 (k=10).

import Data.List (elemIndices)
a045920 n = a045920_list !! (n-1)
a045920_list = map (+ 1) $ elemIndices 0 a076191_list
-- Reinhard Zumkeller, Mar 23 2012, Oct 11 2011

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};Do[If[f[n]==f[n+1],AppendTo[lst,n]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
Transpose[Transpose[Select[Partition[Table[{n,PrimeOmega[n]},{n,400}], 2,1], #[[1,2]]==#[[2,2]]&]][[1]]][[1]] (* Harvey P. Dale, Feb 21 2012 *)
Position[Differences[PrimeOmega[Range[400]]], 0] // Flatten (* Zak Seidov, Oct 30 2012 *)

is(n)=bigomega(n)==bigomega(n+1) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A278291(n) - 1. - Zak Seidov, Nov 17 2018

More terms from David W. Wilson

A093548 a(n) is the smallest number m such that each of the numbers m and m+1 has n distinct prime divisors.

Original entry on oeis.org

2, 14, 230, 7314, 254540, 11243154, 965009045, 65893166030, 5702759516090, 490005293940084, 76622240600506314

Offset: 1

Farideh Firoozbakht, Apr 06 2004

Prime factors may be repeated in m and m+1. The difference between this sequence and A052215 is that in the latter, no prime factor may be repeated. So A052215 imposes more stringent conditions, hence a(n) <= A052215(n). - N. J. A. Sloane, Nov 21 2015
2^63 < a(12) <= 22593106657425552170. - Donovan Johnson, Jan 08 2009
A115186(n) <= a(n) <= A052215(n). - Zak Seidov, Jan 16 2015

			a(5) = 254540 because 254540=2^2*5*11*13*89; 254541=3*7*17*23*31
and 254540 is the smallest number m which each of the numbers m & m+1 has 5 distinct prime divisors.
In contrast, A052215(5) = 378014 > 254540. - _N. J. A. Sloane_, Nov 21 2015

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 230, p. 65, Ellipses, Paris 2008.

Cf. A052215 (another version), A093549, A093550, A115186.

a[n_] := (For[m=1, !(Length[FactorInteger[m]]==n && Length[FactorInteger[m+1]]==n), m++ ];m);Do[Print[a[n]], {n, 7}]
Flatten[Table[SequencePosition[PrimeNu[Range[260000]],{n,n},1],{n,5}],1][[;;,1]] (* To generate more terms, increase the Range and n constants. *) (* Harvey P. Dale, Jun 08 2023 *)

from sympy import primefactors, primorial
def a(n):
  m = primorial(n)
  while True:
    if len(primefactors(m)) == n:
      if len(primefactors(m+1)) == n: return m
      else: m += 2
    else: m += 1
for n in range(1, 6):
  print(a(n), end=", ") # Michael S. Branicky, Feb 14 2021

a[n_] := (For[m=1, !(Length[FactorInteger[m]]==n && Length[FactorInteger[m+1]]==n), m++ ];m)

a(8), a(9) from Martin Fuller, Jan 17 2006

a(10)-a(11) from Donovan Johnson, Jan 08 2009

A052215 a(n) = smallest number m such that m and m+1 are the product of exactly n distinct primes.

Original entry on oeis.org

2, 14, 230, 7314, 378014, 11243154, 965009045, 65893166030, 5702759516090, 605247139068494, 78971815814237709, 22593106657425552170

Offset: 1

Erich Friedman, Jan 29 2000

Prime factors may not be repeated in m and m+1. The difference between this sequence and A093548 is that in the latter, prime factors may be repeated. So the present sequence imposes more stringent conditions than A093548, hence a(n) >= A093548(n). - N. J. A. Sloane, Nov 21 2015
A115186(n) <= A093548(n) <= a(n). - Zak Seidov, Jan 16 2015
2^63 < a(12) <= 22593106657425552170. [Donovan Johnson, Oct 23 2008]
a(12) confirmed to be the upper limit of the range above. - Bert Dobbelaere, Jun 27 2019

			14 and 15 are both the product of 2 primes.
230 is the 3rd entry because we have (230=2*5*23, 231=3*7*11).

Cf. A093548 (another version), A093549, A093550, A115186, A318896.

Subsequence of A005117.

More terms from Naohiro Nomoto, Jul 08 2001
a(7) from Farideh Firoozbakht, Apr 06 2004
a(8)-a(10) from Martin Fuller, Jan 17 2006
a(11) from Donovan Johnson, Oct 23 2008
a(12) from Bert Dobbelaere, Jun 27 2019

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

A154704 a(n) = smallest number k such that k-1 and k+1 both have exactly n prime divisors (counted with multiplicity).

Original entry on oeis.org

4, 5, 19, 55, 271, 1889, 10529, 59777, 101249, 406783, 6581249, 12164095, 65071999, 652963841, 6548416001, 13858918399, 145046192129, 75389157377, 943344975871, 23114453401601, 108772434771967, 101249475018751, 551785225781249, 9740041658826751, 136182187711004671, 4560483868737535

Offset: 1

Klaus Brockhaus, Jan 14 2009, Jan 15 2009

nonn

Similar to A154598, where k is restricted to primes.

m=2*a(n) is the least number m such that m-2 and m+2 have exactly n+1 prime factors, counted with multiplicity. - Hugo Pfoertner, Apr 02 2024

			For k = 4, k-1 = 3 and k+1 = 5 (twin primes) both have one factor and 4 is the smallest such number.
For k = 55, k-1 = 54 = 2*3*3*3 and k+1 = 56 = 2*2*2*7 both have four factors and 55 is the smallest such number.
For k = 59777, k-1 = 59776 = 2*2*2*2*2*2*2*467 and k+1 = 59778 = 2*3*3*3*3*3*3*41 both have eight factors and 59777 is the smallest such number.

David A. Corneth, Table of n, a(n) for n = 1..36
David A. Corneth, Upper bounds on a(n) for n = 1..89

Cf. A115186, A154598.

Cf. A001222 (number of prime divisors of n).

S:=[]; for n:=1 to 10 do k:=3; while not &+[ f[2]: f in Factorization(k-1) ] eq n or not &+[ f[2]: f in Factorization(k+1) ] eq n do k+:=1; end while; Append(~S, k); end for; S;

a[n_]:=Module[{k=2}, While[PrimeOmega[k-1]!=n || PrimeOmega[k+1]!=n, k++]; k]; Array[a,26] (* Stefano Spezia, Apr 02 2024 *)
Flatten[Table[Position[Partition[PrimeOmega[Range[410000]],3,1],?(#[[1]]==#[[3]]==n&),1,1],{n,10}]]+1//Quiet (* The program generates the first ten terms of the sequence. *) (* _Harvey P. Dale, Jul 21 2024 *)
Table[SequencePosition[PrimeOmega[Range[410000]],{n,,n},1],{n,10}][[;;,1,1]]+1 (* The program generates the first ten terms of the sequence. *) (* _Harvey P. Dale, Aug 29 2024 *)

generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(bigomega(m*q+2) == k, listput(list, m*q+1))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Aug 12 2023

a(n) = 2*A115186(n-1) + 1 for n > 1. - Hugo Pfoertner, Apr 02 2024

a(15)-a(23) from Donovan Johnson, Jan 21 2009

a(24)-a(26) from Daniel Suteu, Aug 12 2023

A343818 a(n) is the least number k such that k and k+1 both have n Fermi-Dirac factors (A064547).

Original entry on oeis.org

2, 14, 104, 2079, 21735, 3341624, 103488384, 6110171144

Offset: 1

Amiram Eldar, Apr 30 2021

Since the number of infinitary divisors of k is A037445(k) = 2^A064547(k), a(n) is also the least number k such that k and k+1 both have 2^n infinitary divisors.

a(9) > 2*10^11, if it exists.

			a(1) = 2 since A064547(2) = A064547(3) = 1.
a(2) = 14 since A064547(14) = A064547(15) = 2.

Cf. A064547, A343819.

Similar sequences: A045920, A052215, A075036, A093548, A115186.

fd[1] = 0; fd[n_] := Plus @@ DigitCount[FactorInteger[n][[;;,2]], 2, 1]; seq[m_] := Module[{s = Table[0, {m}], c = 0, n = 1, fd1, fd2}, fd1=fd[n]; While[c < m, fd2 = fd[++n]; If[fd1 == fd2 && fd1 <= m && s[[fd1]] == 0, s[[fd1]] = n-1; c++]; fd1=fd2]; s]; seq[5]

A369897 Numbers k such that k and k + 1 each have 10 prime divisors, counted with multiplicity.

Original entry on oeis.org

3290624, 4122495, 4402431, 5675264, 6608384, 6890624, 7914752, 8614592, 9454400, 11553920, 12613887, 13466816, 14493248, 14853375, 15473024, 16719615, 17494784, 18272384, 18309375, 22784895, 24890624, 25200800, 25869375, 25957503, 26903744, 26921727, 27510272, 28350080, 29761424, 31802624

Offset: 1

Zak Seidov and Robert Israel, Feb 04 2024

nonn

Numbers k such that k and k + 1 are in A046314.

If a and b are coprime terms of A046312, one of them even, then Dickson's conjecture implies there are infinitely many terms k where k/a and (k+1)/b are primes.

			a(5) = 6608384 is a term because 6608384 = 2^9 * 12907 and 6608385 = 3^6 * 5 * 7^2 * 37 each have 10 prime divisors, counted with multiplicity.

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

Cf. A001222, A046312, A046314, A115186.

with(priqueue):
R:= NULL:  count:= 0:
initialize(Q); r:= 0:
insert([-2^10, [2$10]],Q);
while count < 30 do
  T:= extract(Q);
  if -T[1] = r + 1 then
    R:= R, r; count:= count+1;
  fi;
  r:= -T[1];
  p:= T[2][-1];
  q:= nextprime(p);
  for i from 10 to 1 by -1 while T[2][i] = p do
    insert([-r*(q/p)^(11-i), [op(T[2][1..i-1]),q$(11-i)]],Q);
  od
od:
R;

A371651 a(n) is the first prime p such that p - 2 and p + 2 both have exactly n prime factors, counted with multiplicity.

Original entry on oeis.org

5, 23, 173, 2693, 32587, 495637, 4447627, 35303123, 717591877, 928090627, 69692326373, 745041171877, 5012236328123, 64215009765623, 945336806640623, 8885812685546873

Offset: 1

Robert Israel, Apr 01 2024

a(n) is the first prime p such that A001222(p - 2) = A001222(p + 2) = n.
3*10^9 < a(13) <= 5012236328123.
3*10^9 < a(14) <= 64215009765623.

			a(3) = 173 because 173 is prime, 173 - 2 = 171 = 3^2 * 19 and 173 + 2 = 175 = 5^2 * 7 are both products of 3 primes with multiplicity, and no smaller number works.

Cf. A001222. Contained in A371622.

Cf. A115186, A154704.

V:= Vector(8):
p:= 3: count:= 0:
while count < 8 do
p:= nextprime(p);
i:= numtheory:-bigomega(p-2);
if i <= 8 and V[i] = 0 and numtheory:-bigomega(p+2) = i
     then V[i]:= p; count:= count+1
  fi
od:
convert(V,list);

generate(A, B, n) = A=max(A, 2^n); (f(m, p, j) = my(list=List()); if(j==1, forprime(q=max(p,ceil(A/m)), B\m, my(t=m*q); if(isprime(t-2) && bigomega(t-4) == n, listput(list, t-2))), forprime(q = p, sqrtnint(B\m, j), list=concat(list, f(m*q, q, j-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Apr 13 2024

from sympy import primeomega, nextprime
def A371651(n):
    p = 3
    while True:
        if n == primeomega(p-2) == primeomega(p+2):
            return p
        p = nextprime(p) # Chai Wah Wu, Apr 02 2024

a(n) > 2*A154704(n) for n > 1.

a(11) from Michael S. Branicky, Apr 01 2024
a(12) from Michael S. Branicky, Apr 02 2024
a(13) from Chai Wah Wu, Apr 04 2024
a(14)-a(16) from Daniel Suteu, Apr 13 2024

A278294 a(n) = first term in A278291 with n prime factors.

Original entry on oeis.org

3, 10, 28, 136, 945, 5265, 29889, 50625, 203392, 3290625, 6082048, 32536000, 326481921, 3274208001, 6929459200, 72523096065, 37694578689, 471672487936, 11557226700801, 54386217385984, 50624737509376, 275892612890625, 4870020829413376, 68091093855502336, 2280241934368768

Offset: 1

Ely Golden, Nov 16 2016

nonn

a(n)-1 is the first term in A115186 with n prime factors, by definition.

If f(n) is the index of the first occurrence of n in A278293, a(n)=A278291(f(n)), by definition.

			a(2)=10, as it is the first term of A278291 with 2 prime factors.
a(3)=28, as it is the first term of A278291 with 3 prime factors.

Ely Golden, Table of n, a(n) for n = 1..25

Cf. A115186.

import java.math.BigInteger;
public class A278294{
public static void main(String[] args)throws Exception{
    BigInteger dim0=numberOfPrimeFactors(BigInteger.valueOf(2));//note that this method must be manually implemented by the user
    BigInteger dim1;
    BigInteger maxdim=BigInteger.ONE;
    BigInteger counter=BigInteger.valueOf(3);
    long index=1;
    while(index<=20){
      dim1=numberOfPrimeFactors(counter);
      if(dim1.equals(dim0)&&maxdim.testBit(dim1.intValue())==false){System.out.println(dim1+" "+counter);maxdim=maxdim.setBit(dim1.intValue());index++;}
      dim0=dim1;
      counter=counter.add(BigInteger.ONE);
    }
  }
}

Function[w, Flatten@ Map[w[[#]] &, #] &@ Map[First, DeleteCases[#, w_ /; Length@ w == 0]] &@ Map[Position[w, k_ /; PrimeOmega@k == #] &, Range@ 9]]@ Select[Range[10^6], Equal @@ Map[PrimeOmega, {# - 1, #}] &] (* Michael De Vlieger, Dec 01 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);
maxdim=1
counter=3
index=1
while(index<=20):
    dim1=bigomega(counter);
    if((dim1==dim0)&((maxdim&(1<

A330089 a(n) is the smallest integer k such that Omega(k) = n and Omega(2*k+1) = n+1 (where Omega is A001222).

Original entry on oeis.org

7, 22, 148, 472, 2632, 14944, 25312, 349312, 1645312, 16274560, 27565312, 163240960, 1637104000, 17758898176, 36261548032, 18847289344, 1280655450112, 5778613350400, 42629361516544, 219654008209408, 137946306445312, 4629754071040000, 53702255633760256, 9828354353004544, 404693465879805952

Offset: 1

Zak Seidov, Jul 26 2020

nonn

Are all terms > 7 even? Is the sequence infinite?

If A115186(n + 1) is even then a(n) = A115186(n + 1) / 2. - David A. Corneth, Mar 23 2021

			n=1: 7 is prime and 2*7+1 = 15 = 3*5 is semiprime A001358(6);
n=2: 22 = 2*11 is semiprime A001358(8) and 2*22+1 = 45 = 3*3*5 is 3-almost prime A014612(10).

David A. Corneth, Table of n, a(n) for n = 1..27

Cf. A001222, A001358, A014612, A115186.

a(n) = {my(k=1); while ((bigomega(k) != n) || (bigomega(2*k+1) != (n+1)), k++); k;} \\ Michel Marcus, Aug 20 2020

a(14)-a(18) from Dana Jacobsen, Aug 30 2020

More terms from David A. Corneth, Mar 23 2021

cp's OEIS Frontend

A045920 Numbers m such that the factorizations of m..m+1 have the same number of primes (including multiplicities).

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A093548 a(n) is the smallest number m such that each of the numbers m and m+1 has n distinct prime divisors.

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A052215 a(n) = smallest number m such that m and m+1 are the product of exactly n distinct primes.

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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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A154704 a(n) = smallest number k such that k-1 and k+1 both have exactly n prime divisors (counted with multiplicity).

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Magma

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A343818 a(n) is the least number k such that k and k+1 both have n Fermi-Dirac factors (A064547).

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Mathematica

A369897 Numbers k such that k and k + 1 each have 10 prime divisors, counted with multiplicity.

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A371651 a(n) is the first prime p such that p - 2 and p + 2 both have exactly n prime factors, counted with multiplicity.