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

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

Offset: 1

Author

Farideh Firoozbakht, Apr 06 2004

Keywords

• nonn
• more

Comments

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

Examples

			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
		

References

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

Crossrefs

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

Programs

Mathematica

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 *)

Python

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

Formula

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

Extensions

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

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