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A051250
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Numbers whose reduced residue system consists of 1 and prime powers only.
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9
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1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 18, 20, 24, 30, 42, 60
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OFFSET
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1,2
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COMMENTS
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Conjecture: the sequence is finite and 60 is the largest term, empirically verified up to 10^7;
The sequence is indeed finite. Let pi*(x) denote the number of prime powers (including 1) up to x. Dusart's bounds plus finite checking [up to 60184] shows that pi*(x) <= x/(log(x) - 1.1) + sqrt(x) for x >= 4. phi(n) > n/(e^gamma log log n + 3/(log log n)) for n >= 3. Convexity plus finite checking [up to 1096] allows a quick proof that phi(n) > pi*(n) for n > 420. So if n > 420, the reduced residue system mod n must contain at least one number that is neither 1 nor a prime power. Hence 60 is the last term in the sequence. - Charles R Greathouse IV, Jul 14 2011
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LINKS
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O. Ore and N. J. Fine, Reduced Residue Systems, American Mathematical Monthly Vol. 66, No. 10 (Dec., 1959), pp. 926-927.
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EXAMPLE
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RRS[ 60 ] = {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}.
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MATHEMATICA
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fQ[n_] := Union[# == 1 || Mod[#, # - EulerPhi[#]] == 0 & /@ Select[ Range@ n, GCD[#, n] == 1 &]] == {True}; Select[ Range@ 100, fQ] (* Robert G. Wilson v, Jul 11 2011 *)
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PROG
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(Haskell)
a051250 n = a051250_list !! (n-1)
a051250_list = filter (all ((== 1) . a010055) . a038566_row) [1..]
(PARI) isprimepower(n)=ispower(n, , &n); isprime(n)
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CROSSREFS
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KEYWORD
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nice,nonn,fini,full
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AUTHOR
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STATUS
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approved
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