A066169 Least k such that phi(k) >= n.

1, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 31, 31, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 43, 43, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59, 59, 61, 61, 67, 67, 67, 67, 67, 67, 71, 71, 71, 71, 73

Offset: 1

Author

Joseph L. Pe, Dec 13 2001

Keywords

• nonn
• easy

Comments

Thinking of n as time, a(n) represents the first time phi catches up with i(n), where i is the identity function. a(n) - n can be seen as the lag of phi behind i at time n. The sequence of these lags begins 0 1 2 1 2 1 4 3 2 1 2 1 4 3 2 1 2 1 4 3 2 1

a(n) is the smallest number for which the reduced residue system (=RRS(a(n))) contains {1,2,...,n} as a subset; a(m) jumps at a(p)-1 and a(p) from value of p to nextprime(p); a(x)=p(n) holds {p(n-1)...p(n)-1}; p(n) is repeated p(n)-p(n-1) times. For n > 1, a(n) = p(Pi(n)+1), while a(1)=1. - Labos Elemer, May 14 2003

Examples

			a(5) = 7 since phi(7) = 6 is at least 5 and 7 is the smallest k satisfying phi(k) is greater than or equal to 5.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A000720, A057237.

Programs

Mathematica

a(1)=1; Table[Prime[PrimePi[w]+1], {w, 1, 100}]

PARI

{ for (n=1, 1000, k=1; while (eulerphi(k) < n, k++); write("b066169.txt", n, " ", k) ) } \\ Harry J. Smith, Feb 04 2010

PARI

print1(n=1);n=2;forprime(p=3,31,while(n++<=p,print1(", "p));n--) \\ Charles R Greathouse IV, Oct 31 2011

Formula

a(1) = 1 a(n) = p(s+1) for n in [p(s), p(s+1) - 1], where p(s) denotes the s-th prime.

For n > 1 a(n) = A007918(n+1). - Benoit Cloitre, May 04 2002

For n > 1, a(n) = A000040(A000720(n)+1), while a(1)=1. - Labos Elemer, May 14 2003

Extensions

More terms from Benoit Cloitre, May 04 2002