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 A067133 n is a term if the phi(n) numbers in [0,n-1] and coprime to n form an arithmetic progression. 1
 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; refs; listen; history; text; internal format)
 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/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. From Bernard Schott, Jan 08 2021: (Start) This sequence is the answer to the 2nd problem, proposed by Romania, during the 32nd International Mathematical Olympiad in 1991 at Sigtuna (Sweden) (see the link IMO Compendium and reference Kuczma). These phi(m) numbers coprimes to m form an arithmetic progression with at least 3 terms iff m = 5 or m >= 7. (End) REFERENCES Marcin E. Kuczma, International Mathematical Olympiads, 1986-1999, The Mathematical Association of America, 2003, pages 6 and 61-62. LINKS Table of n, a(n) for n=1..61. The IMO Compendium, Problem 2, 32nd IMO 1991. Index to sequences related to Olympiads. 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, n-1 ], GCD[ #, n ]==1& ]; difs[ n_ ] := Drop[ n, 1 ]-Drop[ n, -1 ]; Select[ Range[ 1, 250 ], Length[ Union[ difs[ rps[ # ] ] ] ]<=1& ] PROG (PARI) isok(n) = {my(v = select(x->gcd(x, n)==1, [1..n]), dv = vector(#v-1, k, v[k+1] - v[k])); if (#dv, if (vecmin(dv) != vecmax(dv), return(0))); return(1)} \\ Michel Marcus, Jan 08 2021 CROSSREFS Equals A000040 U A000079 U {6}. Equals A174090 U {6}. Sequence in context: A084369 A167211 A362095 * A192588 A351914 A258946 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

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Last modified September 12 11:46 EDT 2024. Contains 375851 sequences. (Running on oeis4.)