

A067133


n is a term if the phi(n) numbers in [0,n1] 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/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.
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, 19861999, The Mathematical Association of America, 2003, pages 6 and 6162.


LINKS

The IMO Compendium, Problem 2, 32nd IMO 1991.


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& ]


PROG

(PARI) isok(n) = {my(v = select(x>gcd(x, n)==1, [1..n]), dv = vector(#v1, k, v[k+1]  v[k])); if (#dv, if (vecmin(dv) != vecmax(dv), return(0))); return(1)} \\ Michel Marcus, Jan 08 2021


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



