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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.

%I A093548 #24 Jun 08 2023 12:34:19
%S A093548 2,14,230,7314,254540,11243154,965009045,65893166030,5702759516090,
%T A093548 490005293940084,76622240600506314
%N A093548 a(n) is the smallest number m such that each of the numbers m and m+1 has n distinct prime divisors.
%C A093548 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
%C A093548 2^63 < a(12) <= 22593106657425552170. - _Donovan Johnson_, Jan 08 2009
%C A093548 A115186(n) <= a(n) <= A052215(n). - _Zak Seidov_, Jan 16 2015
%D A093548 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 230, p. 65, Ellipses, Paris 2008.
%F A093548 a[n_] := (For[m=1, !(Length[FactorInteger[m]]==n && Length[FactorInteger[m+1]]==n), m++ ];m)
%e A093548 a(5) = 254540 because 254540=2^2*5*11*13*89; 254541=3*7*17*23*31
%e A093548 and 254540 is the smallest number m which each of the numbers m & m+1 has 5 distinct prime divisors.
%e A093548 In contrast, A052215(5) = 378014 > 254540. - _N. J. A. Sloane_, Nov 21 2015
%t A093548 a[n_] := (For[m=1, !(Length[FactorInteger[m]]==n && Length[FactorInteger[m+1]]==n), m++ ];m);Do[Print[a[n]], {n, 7}]
%t A093548 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 *)
%o A093548 (Python)
%o A093548 from sympy import primefactors, primorial
%o A093548 def a(n):
%o A093548   m = primorial(n)
%o A093548   while True:
%o A093548     if len(primefactors(m)) == n:
%o A093548       if len(primefactors(m+1)) == n: return m
%o A093548       else: m += 2
%o A093548     else: m += 1
%o A093548 for n in range(1, 6):
%o A093548   print(a(n), end=", ") # _Michael S. Branicky_, Feb 14 2021
%Y A093548 Cf. A052215 (another version), A093549, A093550, A115186.
%K A093548 nonn,more
%O A093548 1,1
%A A093548 _Farideh Firoozbakht_, Apr 06 2004
%E A093548 a(8), a(9) from _Martin Fuller_, Jan 17 2006
%E A093548 a(10)-a(11) from _Donovan Johnson_, Jan 08 2009