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Search: id:A120314
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| A120314 |
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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). |
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| 2, 9, 81450625, 6080399213078595601, 38604666779024731098340977806401, 79600343456925208350554324952070658488321, 111999530649584986702170994086297063568244097100801
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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.
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FORMULA
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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).
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EXAMPLE
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Simplest: a(2)=9=3^2 since tau(9)=3; 9-1=8=2^3 so tau(8)=4.
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MAPLE
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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;
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CROSSREFS
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Cf. A033950, A036898, A114617.
Sequence in context: A132859 A103562 A140319 this_sequence A050924 A096877 A058297
Adjacent sequences: A120311 A120312 A120313 this_sequence A120315 A120316 A120317
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KEYWORD
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nonn
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AUTHOR
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Walter Kehowski (wkehowski(AT)cox.net), Jun 20 2006
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