A002174 Values taken by reduced totient function psi(n).

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 90, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

If p is a Sophie Germain prime (A005384), then 2p is here. - T. D. Noe, Aug 13 2008

Terms of A002322, sorted and multiple values taken just once. - Vladimir Joseph Stephan Orlovsky, Jul 21 2009

a(2445343) = 10^7, suggesting that Luca & Pomerance's lower bound may be closer to the truth than the upper bound. The fit exponent log a(n)/log n - 1 = 0.0957... in this case. - Charles R Greathouse IV, Jul 02 2017

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• D. H. Lehmer, Guide to Tables in the Theory of Numbers, Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
• Florian Luca and Carl Pomerance, On the range of Carmichael's universal-exponent function, Acta Arithmetica 162 (2014), pp.289-308.
• C. Moreau, Sur quelques théorèmes d'arithmétique, Nouvelles Annales de Mathématiques, 17 (1898), 293-307.

Crossrefs

Cf. A002322, A002396, A143407, A143408.

Programs

Mathematica

lst={}; Do[AppendTo[lst, CarmichaelLambda[n]], {n, 6*7!}]; lst; Take[Union[lst], 123] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
(* warning: there seems to be no guarantee that no terms near the end are omitted! - Joerg Arndt, Dec 23 2014 *)
TakeWhile[Union@ Table[CarmichaelLambda@ n, {n, 10^6}], # <= 168 &] (* Michael De Vlieger, Mar 19 2016 *)

PARI

list(lim)=my(v=List([1]),u,t); forprime(p=3,lim\3+1, u=List(); listput(u,p-1); while((t=u[#u]*p)<=lim, listput(u,t)); for(j=1,#v, for(i=1,#u, t=lcm(u[i],v[j]); if(t<=lim && t!=v[j], listput(v,t)))); v=List(Set(v))); forprime(p=lim\3+2,lim+1, listput(v,p-1)); v=List(Set(v)); for(i=1,#v, t=2*v[i]; if(t>lim, break); listput(v,t); while((t*=2)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jun 23 2017

PARI

is(n)=if(n%2, return(n==1)); my(f=factor(n),pe); for(i=1,#f~, if(n%(f[i,1]-1)==0, next); pe=f[i,1]^f[i,2]; forstep(q=2*pe+1,n+1,2*pe, if(n%(q-1)==0 && isprime(q), next(2))); return(0)); 1 \\ Charles R Greathouse IV, Jun 25 2017

Formula

n (log n)^0.086 << a(n) << n (log n)^0.36 where << is the Vinogradov symbol, see Luca & Pomerance. - Charles R Greathouse IV, Dec 28 2013

Extensions

More terms from T. D. Noe, Aug 13 2008