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 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). 0
 2, 9, 81450625, 6080399213078595601, 38604666779024731098340977806401, 79600343456925208350554324952070658488321, 111999530649584986702170994086297063568244097100801 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Table of n, a(n) for n=1..7. FORMULA 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). EXAMPLE Simplest: a(2)=9=3^2 since tau(9)=3; 9-1=8=2^3 so tau(8)=4. MAPLE 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; CROSSREFS Cf. A033950, A036898, A114617. Sequence in context: A103562 A309861 A140319 * A050924 A181500 A189876 Adjacent sequences: A120311 A120312 A120313 * A120315 A120316 A120317 KEYWORD nonn AUTHOR Walter Kehowski, Jun 20 2006 STATUS approved

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Last modified July 21 00:40 EDT 2024. Contains 374462 sequences. (Running on oeis4.)