A052215 -id:A052215 - cp's OEIS frontend

A006049 Numbers k such that k and k+1 have the same number of distinct prime divisors.

2, 3, 4, 7, 8, 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, 91, 92, 93, 94, 95, 98, 99, 111, 115, 116, 117, 118, 122, 123, 127, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 152, 158, 159, 160, 161, 171, 175

Offset: 1

N. J. A. Sloane

Sequence is infinite, as proved by Schlage-Puchta, who comments: "Buttkewitz found a non-computational proof, and the Goldston-Pintz-Yildirim-sieve yields more precise information". - Charles R Greathouse IV, Jan 09 2013

The asymptotic density of this sequence is 0 (Erdős, 1936). - Amiram Eldar, Sep 17 2024

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from T. D. Noe)
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.
Jan-Christoph Schlage-Puchta, The equation ω(n)=ω(n+1), Mathematika, Vol. 50, No. 1-2 (2003), pp. 99-101; arXiv preprint, arXiv:1105.1621 [math.NT], 2011.

Cf. A001221, A052215, A107800, A006073, A294277, A294278.

Subsequence of A062974.

import Data.List (elemIndices)
a006049 n = a006049_list !! (n-1)
a006049_list = map (+ 1) $ elemIndices 0 $
   zipWith (-) (tail a001221_list) a001221_list
-- Reinhard Zumkeller, Jan 22 2013

f[n_] := Length@FactorInteger[n];t = f /@ Range[175];Flatten@Position[Rest[t] - Most[t], 0] (* Ray Chandler, Mar 27 2007 *)
Select[Range[200],PrimeNu[#]==PrimeNu[#+1]&] (* Harvey P. Dale, May 09 2012 *)
Flatten[Position[Partition[PrimeNu[Range[200]],2,1],?(#[[1]]==#[[2]]&),{1},Heads->False]] (* _Harvey P. Dale, May 22 2015 *)
SequencePosition[PrimeNu[Range[200]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 02 2019 *)

is(n)=omega(n)==omega(n+1) \\ Charles R Greathouse IV, Jan 09 2013

A001221(a(n)) = A001221(a(n)+1). - Reinhard Zumkeller, Jan 22 2013

Extended by Ray Chandler, Mar 27 2007

A115186 Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

2, 9, 27, 135, 944, 5264, 29888, 50624, 203391, 3290624, 6082047, 32535999, 326481920, 3274208000, 6929459199, 72523096064, 37694578688, 471672487935, 11557226700800, 54386217385983, 50624737509375, 275892612890624, 4870020829413375, 68091093855502335, 2280241934368767, 809386931759611904, 519017301463269375

Offset: 1

Reinhard Zumkeller, Jan 16 2006

nonn

A001222(a(n)) = A001222(a(n)+1) = n: subsequence of A045920.
a(16) > 4*10^10. - Martin Fuller, Jan 17 2006
a(n) <= A093548(n) <= A052215(n). - Zak Seidov, Jan 16 2015
Apparently, 4*a(n)+2 is the least number k such that k-2 and k+2 have exactly n+2 prime factors, counted with multiplicity. - Hugo Pfoertner, Apr 02 2024

			a(10) = 3290624 = 6427 * 2^9, 3290624+1 = 13 * 5^5 * 3^4:
A001222(3290624) = A001222(3290625) = 10.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 135, p. 46, Ellipses, Paris 2008.

Cf. A001222, A045920, A052215, A093548.

Equivalent sequences for longer runs: A113752 (3), A356893 (4).

f:= proc(n) uses priqueue; local t,x,p,i;
    initialize(pq);
    insert([-3^n, 3$n], pq);
    do
      t:= extract(pq);
      x:= -t[1];
      if numtheory:-bigomega(x-1)=n then return x-1
      elif numtheory:-bigomega(x+1)=n then return x
      fi;
      p:= nextprime(t[-1]);
      for i from n+1 to 2 by -1 while t[i] = t[-1] do
        insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
      od;
    od
end proc:
seq(f(i),i=1..27); # Robert Israel, Sep 30 2024

a(13)-a(15) from Martin Fuller, Jan 17 2006
a(16)-a(17) from Donovan Johnson, Apr 08 2008
a(18)-a(22) from Donovan Johnson, Jan 21 2009
a(23)-a(25) from Donovan Johnson, May 25 2013
a(26)-a(27) from Robert Israel, Sep 30 2024

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

A093550 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 is a product of n distinct primes.

Original entry on oeis.org

34, 1310, 203434, 16467034, 1990586014, 41704979954, 102099792179230, 22192526378762466

Offset: 2

Farideh Firoozbakht, Apr 07 2004, corrected Aug 26 2006

Each term of this sequence is of the form 4k+2.

			a(5)=16467034 because each of the three numbers 16467034-1, 16467034 & 16467034+1 are products of 5 distinct primes (16467033=3*11*17*149*197, 16467034=2*19*23*83*227, 16467035=5*13*37*41*167) and 16467034 is the smallest such number.

Jacques Tramu, Puzzle 371

Cf. A093549, A052215, A093548, A248201, A248202, A248203, A248204, A259349, A259350, A259801.

a[n_] := a[n] = (For[m=1, !(Length[FactorInteger[4m+1]]==n && SquareFreeQ[4m+1] && Length[FactorInteger[4m+2]]==n && SquareFreeQ[4m+2] && Length[FactorInteger[4m+3]]==n && SquareFreeQ[4m+3]), m++ ]; 4m+2); Table[Print[a[n]]; a[n], {n, 2, 6}] (* updated by Jean-François Alcover, Jul 04 2013 *)

a(n)={my(m=1);while(!(issquarefree(m-1)&&issquarefree(m)&&issquarefree(m+1)&&omega(m-1)==n&&omega(m)==n&&omega(m+1)==n),m++);return(m);} main(size)={my(n);return(vector(size,n,a(n+1)));} /* Anders Hellström, Jul 14 2015 */

a(7) added from Jacques Tramu's web site by Farideh Firoozbakht, Aug 26 2006
a(8) from Donovan Johnson, Oct 27 2008
a(9) from James G. Merickel, Jul 24 2015

A318896 Numbers k such that k and k+1 are the product of exactly four distinct primes.

Original entry on oeis.org

7314, 8294, 8645, 11570, 13629, 13845, 15105, 15554, 16554, 17390, 17654, 18290, 19005, 20405, 20769, 21489, 22010, 22154, 23001, 23114, 23529, 24530, 24765, 24870, 24969, 25346, 26690, 26894, 26961, 27434, 27965, 28105, 29145, 29210, 29414, 29469, 29666, 30414

Offset: 1

Seiichi Manyama, Sep 05 2018

nonn

This sequence is different from A140078. For example, A140078(4) = 9009 = 3^2 * 7 * 11 * 13 is not a term.

			n | a(n)                    | a(n)+1
--+-------------------------+-------------------------
1 | 7314 = 2 *  3 * 23 * 53 | 7315 = 5 * 7 * 11 *  19
2 | 8294 = 2 * 11 * 13 * 29 | 8295 = 3 * 5 *  7 *  79
3 | 8645 = 5 *  7 * 13 * 19 | 8646 = 2 * 3 * 11 * 131

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

Subsequence of A140078.

Cf. A046386, A052215, A215217, A263990, A318964.

is(n) = omega(n)==4 && omega(n+1)==4 && bigomega(n)==4 && bigomega(n+1)==4 \\ Felix Fröhlich, Sep 05 2018

is(n) = factor(n)[, 2]~ == [1, 1, 1, 1] && factor(n+1)[, 2]~ == [1, 1, 1, 1] \\ David A. Corneth, Sep 06 2018

A318964 Numbers k such that both k and k+1 are the product of exactly five distinct primes.

Original entry on oeis.org

378014, 421134, 483405, 486590, 486794, 489345, 507129, 545258, 549185, 558789, 558830, 634809, 637329, 663585, 667029, 690234, 720290, 776985, 782690, 823745, 824109, 853005, 853034, 855645, 873885, 883245, 892905, 935714, 945230, 968253, 987734, 999005, 1005081, 1013726

Offset: 1

Seiichi Manyama, Sep 06 2018

nonn

			n | a(n)                           | a(n)+1
--+--------------------------------+--------------------------------
1 | 378014 = 2 * 7 * 13 * 31 *  67 | 378015 = 3 *  5 * 11 * 29 * 79
2 | 421134 = 2 * 3 *  7 * 37 * 271 | 421135 = 5 * 11 * 13 * 19 * 31
3 | 483405 = 3 * 5 * 13 * 37 *  67 | 483406 = 2 *  7 * 11 * 43 * 73

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Subsequence of A140079.

Cf. A046387, A052215, A215217, A263990, A318896.

is(n) = omega(n)==5 && omega(n+1)==5 && bigomega(n)==5 && bigomega(n+1)==5 \\ Felix Fröhlich, Sep 06 2018

A093549 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 has n distinct prime divisors.

Original entry on oeis.org

3, 21, 645, 37961, 1042405, 323567035, 30989984675, 10042712381261

Offset: 1

Farideh Firoozbakht, Apr 07 2004

a(n) <= A093550(n) since here the factors do not occur necessarily to the first power, e.g. a(2)-1 = 20 = 2^2*5, therefore A093550(2) > a(2). - M. F. Hasler, May 20 2014

			a(3)=645 because 644=2^2*7*23; 645=3*5*43; 646=2*17*19 and 645 is the smallest number m such that each of the numbers m-1, m and m+1 has 3 distinct prime divisors.

Cf. A093550, A052215, A093548.

a[n_] := (For[m=2, !(Length[FactorInteger[m-1]]==n && Length[FactorInteger[m]]==n&&Length[FactorInteger[m+1]]==n), m++ ];m);Do[Print[a[n]], {n, 7}]

a(n,m=2)=until(,for(k=-1,1,omega(m-k)!=n&&(m+=2-k)&&next(2));return(m)) \\ M. F. Hasler, May 20 2014

a[n_] := (For[m=2, !(Length[FactorInteger[m-1]]==n && Length[FactorInteger[m]]==n&&Length[FactorInteger[m+1]]==n), m++ ];m)

a(7) from Donovan Johnson, Apr 07 2008

a(8) from Donovan Johnson, Jan 15 2009

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

Original entry on oeis.org

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

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]

A358818 a(n) is the least number k such that A046660(k) = A046660(k+1) = n.

Original entry on oeis.org

1, 44, 135, 80, 8991, 29888, 123200, 2316032, 1043199, 24151040, 217713663, 689278976, 11573190656, 76876660736, 311969153024, 2035980763136, 2741258240000, 215189482110975

Offset: 0

Amiram Eldar, Dec 02 2022

a(14) <= 314944159743.

a(18) > 10^14.5; a(19) = 275892612890624; a(20) > 10^14.5. - Martin Ehrenstein, Dec 11 2022

Cf. A046660.
Subsequence of A358817.
Similar sequences: A052215, A059737, A093548, A115186.

e[n_] := PrimeOmega[n] - PrimeNu[n]; a[n_] := Module[{k = 1}, While[e[k] != n || e[k + 1] != n, k++]; k]; Array[a, 10, 0]

e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
a(n) = {my(k=1); while(e(k) != n || e(k+1) !=n , k++); k};

a(14)-a(16) from Martin Ehrenstein, Dec 04 2022

a(17) from Martin Ehrenstein, Dec 09 2022

cp's OEIS Frontend

A006049 Numbers k such that k and k+1 have the same number of distinct prime divisors.

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A115186 Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).

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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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A093550 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 is a product of n distinct primes.

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A318896 Numbers k such that k and k+1 are the product of exactly four distinct primes.

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A318964 Numbers k such that both k and k+1 are the product of exactly five distinct primes.

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A093549 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 has n distinct prime divisors.

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