A051250 Numbers whose reduced residue system consists of 1 and prime powers only.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 18, 20, 24, 30, 42, 60

Offset: 1

Author

Labos Elemer

Keywords

• nice
• nonn
• fini
• full

Comments

From Reinhard Zumkeller, Oct 27 2010: (Start)

Conjecture: the sequence is finite and 60 is the largest term, empirically verified up to 10^7;

A139555(a(n)) = A000010(a(n)). (End)

The sequence is indeed finite. Let pi*(x) denote the number of prime powers (including 1) up to x. Dusart's bounds plus finite checking [up to 60184] shows that pi*(x) <= x/(log(x) - 1.1) + sqrt(x) for x >= 4. phi(n) > n/(e^gamma log log n + 3/(log log n)) for n >= 3. Convexity plus finite checking [up to 1096] allows a quick proof that phi(n) > pi*(n) for n > 420. So if n > 420, the reduced residue system mod n must contain at least one number that is neither 1 nor a prime power. Hence 60 is the last term in the sequence. - Charles R Greathouse IV, Jul 14 2011

Examples

			RRS[ 60 ] = {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}.
		

Links

• M. Dalezman, From 30 to 60 Is Not Twice as Hard, Mathematics Magazine, Vol. 73, No. 2 (Apr. 2000), pp. 151-153.
• O. Ore and N. J. Fine, Reduced Residue Systems, American Mathematical Monthly Vol. 66, No. 10 (Dec., 1959), pp. 926-927.

Crossrefs

Cf. A048597, A048862-A048869.

Cf. A010055, A038566.

Programs

Haskell

a051250 n = a051250_list !! (n-1)
a051250_list = filter (all ((== 1) . a010055) . a038566_row) [1..]
-- Reinhard Zumkeller, May 27 2015, Dec 18 2011, Oct 27 2010

Mathematica

fQ[n_] := Union[# == 1 || Mod[#, # - EulerPhi[#]] == 0 & /@ Select[ Range@ n, GCD[#, n] == 1 &]] == {True}; Select[ Range@ 100, fQ] (* Robert G. Wilson v, Jul 11 2011 *)

PARI

isprimepower(n)=ispower(n,,&n);isprime(n)
is(n)=for(k=2,n-1,if(gcd(n,k)==1&&!isprimepower(k),return(0)));1 \\ Charles R Greathouse IV, Jul 14 2011