A120314 - OEIS

%I #3 Mar 30 2012 18:36:04

%S 2,9,81450625,6080399213078595601,38604666779024731098340977806401,

%T 79600343456925208350554324952070658488321,

%U 111999530649584986702170994086297063568244097100801

%N 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 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 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 Simplest: a(2)=9=3^2 since tau(9)=3; 9-1=8=2^3 so tau(8)=4.

%p 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 Cf. A033950, A036898, A114617.

%K nonn

%O 1,1

%A _Walter Kehowski_, Jun 20 2006