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Search: id:A067133
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| A067133 |
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n is a term of this sequence if the phi(n) numbers in [0,n-1] and coprime to n form an arithmetic progression. |
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+0
1
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| 1, 2, 3, 4, 5, 6, 7, 8, 11, 13, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The sequence consists of primes, powers of 2 and 6. Sketch of proof: Let k be the common difference of the arithmetic progression. If n is odd, then 1 and 2 are coprime to n, so k=1 and n is prime. If n==0 (mod 4), then n/2-1 and n/2+1 are coprime to n, so k=2 and n is a power of 2. If n==2 (mod 4), then n/2-2 and n/2+2 are coprime to n, so k divides 4 and n is either 2 or 6.
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EXAMPLE
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8 is a term as phi(8) = 4 and the coprime numbers 1,3,5,7 form an arithmetic progression. 17 is a member as phi(17) = 16 and the numbers 1 to 16 form an arithmetic progression.
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MATHEMATICA
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rps[ n_ ] := Select[ Range[ 0, n-1 ], GCD[ #, n ]==1& ]; difs[ n_ ] := Drop[ n, 1 ]-Drop[ n, -1 ]; Select[ Range[ 1, 250 ], Length[ Union[ difs[ rps[ # ] ] ] ]<=1& ]
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CROSSREFS
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Sequence in context: A093515 A084369 A167211 this_sequence A008815 A118600 A080543
Adjacent sequences: A067130 A067131 A067132 this_sequence A067134 A067135 A067136
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KEYWORD
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easy,nonn
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 09 2002
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EXTENSIONS
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Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Jan 15 2002
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