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Search: id:A070159
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| A070159 |
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Numbers n such that phi[n]/(sigma[n]-n) is an integer. |
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3
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| 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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This sequence consists of all primes p (for which the given ratio equals (p-1)/1, see A000040) and of composites listed in A055940 (see examples).
Up to 10^7, there is no element of this sequence having more than 2 prime factors. - M. F. Hasler, Dec 11 2007
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FORMULA
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{ a(k) } = { n in N | A000010(n)/A001065(n) is an integer }.
{ a(k) } = { A000040(k) } union { A055940(k) }.
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EXAMPLE
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The prime p=47 is in this sequence since phi[p]/(sigma[p]-p) = p-1 is an integer, as is the case for any other prime.
The composite n=403=13*31 is in this sequence, since the ratio Phi[n]/(sigma[n]-n) =360/(1+13+31)=8 is an integer.
The first few composites in this sequence are 133,403,583,713,... (A055940)
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MATHEMATICA
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Do[s=EulerPhi[n]/(DivisorSigma[1, n]-n); If[IntegerQ[s], Print[n]], {n, 2, 1000}]
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PROGRAM
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(PARI) for(n=2, 999, eulerphi(n)%(sigma(n)-n)|print1(n", ")) \\ - M. F. Hasler, Dec 11 2007
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CROSSREFS
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Cf. A000010, A000040, A001065, A000203, A055940, A070037, A020492, A068418, A062972.
Sequence in context: A137589 A052424 A055398 this_sequence A158611 A000040 A008578
Adjacent sequences: A070156 A070157 A070158 this_sequence A070160 A070161 A070162
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KEYWORD
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nonn
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Apr 26 2002
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EXTENSIONS
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Edited by M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Dec 11 2007
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