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A070162
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Numbers k such that k - phi(k) - 1 is a prime.
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2
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6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 26, 34, 36, 38, 40, 42, 44, 46, 48, 50, 56, 58, 60, 62, 64, 72, 74, 78, 80, 82, 84, 86, 88, 92, 94, 100, 106, 108, 116, 118, 122, 126, 134, 136, 142, 146, 150, 152, 156, 158, 162, 164, 166, 178, 180, 182, 192, 194, 198, 202, 204
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OFFSET
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1,1
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COMMENTS
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Maximal solution is either k=2p or, in the case of Mersenne primes M, k=2(M+1) and f(k)=p or f(k)=M.
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LINKS
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FORMULA
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EXAMPLE
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k = 192: phi(192) = 64, cototient(192) = 128, k - phi(192) - 1 = 127 is prime;
k = 2p: 2p - phi(2p) - 1 = 2p - p + 1 - 1 = p, so 2*prime is always a solution;
k = 2^(q+1), where q is a Mersenne prime exponent: cototient(k) - 1 = 2^(p+1) - 2^p - 1 = 2^p - 1, which is the corresponding Mersenne prime.
Numbers k in {192, 224, 248, 254, 256} give p = 127;
numbers k in {72, 80, 88, 92, 94} give p = 47.
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MATHEMATICA
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Do[s=n-EulerPhi[n]-1; If[PrimeQ[s], Print[n, s]], n, 1, 10000]
Select[Range[250], PrimeQ[#-EulerPhi[#]-1]&] (* Harvey P. Dale, Sep 05 2023 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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