cp's OEIS Frontend

A120314 First occurrence of consecutive refactorable numbers a(n)-1, a(n) where the smallest prime factor of a(n) is the n-th prime prime(n).

%I A120314 #3 Mar 30 2012 18:36:04
%S A120314 2,9,81450625,6080399213078595601,38604666779024731098340977806401,
%T A120314 79600343456925208350554324952070658488321,
%U A120314 111999530649584986702170994086297063568244097100801
%N A120314 First occurrence of consecutive refactorable numbers a(n)-1, a(n) where the smallest prime factor of a(n) is the n-th prime prime(n).
%C A120314 The factorizations are 2, 3^2, (5^4)*(19^4), (7^6)*(193^6), (11^10)*(131^10), (13^12)*(197^12), (17^16)*(79^16). Since the squares of all odd primes greater than 3 are 1 mod 12, the refactorable a(n)-1 is divisible by 12 whenever n>=3.
%F A120314 a(n) is the first integer of the form (k*p)^(p-1) for some k such that a(n)-1 and a(n) are refactorable and the smallest prime divisor of a(n) is prime(n).
%e A120314 Simplest: a(2)=9=3^2 since tau(9)=3; 9-1=8=2^3 so tau(8)=4.
%p A120314 with(numtheory); RFC:=[[1,2,2]]: for w to 1 do for i from 2 to 12 do p:=ithprime(i); P:=[seq(ithprime(j),j=1..i-1)]; for k from 1 to 12^6 do if andmap(z -> k mod z <>0, P) then n:=(p*k)^(p-1); t:=tau(n); n1:=n-1; t1:=tau(n-1); if (n mod t = 0) and (n1 mod t1 = 0) then RFC:=[op(RFC),[k,p,n]]; print(ifactor(n)); break; fi fi od od od;
%Y A120314 Cf. A033950, A036898, A114617.
%K A120314 nonn
%O A120314 1,1
%A A120314 _Walter Kehowski_, Jun 20 2006