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A241196
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Primes p at which phi(p-1)/(p-1) reaches a new minimum, where phi is Euler's totient function.
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4
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2, 3, 7, 31, 211, 2311, 43891, 78541, 120121, 870871, 1381381, 2282281, 4084081, 13123111, 82192111, 106696591, 300690391, 562582021, 892371481, 6915878971, 71166625531, 200560490131
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OFFSET
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1,1
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COMMENTS
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For these p, the numerator and denominator of phi(p-1)/(p-1) are listed in A241197 and A241198. This sequence appears to be related to A073918, the smallest prime which is 1 more than a product of n distinct primes.
By Dirichlet's theorem on primes in arithmetic progressions, for any n there is a prime p such that p-1 is divisible by the primorial A002110(n). Then phi(p-1)/(p-1) <= Product_{i=1..n} (1 - 1/prime(i)). Since Sum_{i >= 1} prime(i) diverges, that goes to 0 as n -> infinity. Thus there are primes with phi(p-1)/(p-1) arbitrarily close to 0. - Robert Israel, Jan 18 2016
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, A2.
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LINKS
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MAPLE
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m:= infinity:
p:= 1:
count:= 0:
while count < 10 do
p:= nextprime(p);
r:= numtheory:-phi(p-1)/(p-1);
if r < m then
count:= count+1;
A[count]:= p;
m:= r;
fi
od:
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MATHEMATICA
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tMin = {{2, 1}}; Do[p = Prime[n]; tn = EulerPhi[p - 1]/(p - 1); If[tn < tMin[[-1, -1]], AppendTo[tMin, {p, tn}]], {n, 10^7}]; Transpose[tMin][[1]]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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