

A067133


n is a term of this sequence if the phi(n) numbers in [0,n1] and coprime to n form an arithmetic progression.


0



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
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OFFSET

1,2


COMMENTS

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/21 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/22 and n/2+2 are coprime to n, so k divides 4 and n is either 2 or 6.


LINKS

Table of n, a(n) for n=1..61.


EXAMPLE

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.


MATHEMATICA

rps[ n_ ] := Select[ Range[ 0, n1 ], GCD[ #, n ]==1& ]; difs[ n_ ] := Drop[ n, 1 ]Drop[ n, 1 ]; Select[ Range[ 1, 250 ], Length[ Union[ difs[ rps[ # ] ] ] ]<=1& ]


CROSSREFS

Sequence in context: A249724 A084369 A167211 * A192588 A258946 A008815
Adjacent sequences: A067130 A067131 A067132 * A067134 A067135 A067136


KEYWORD

easy,nonn


AUTHOR

Amarnath Murthy, Jan 09 2002


EXTENSIONS

Edited by Dean Hickerson, Jan 15 2002


STATUS

approved



