A105261 Values of n such that phi(n)=c(n)^2, where phi is the Euler totient function and c(n) is the product of the distinct prime factors of n (c(1)=1).

1, 8, 108, 250, 6174, 41154

Offset: 1

Author

Emeric Deutsch, Apr 14 2005

Keywords

• fini
• nonn
• full

Comments

This sequence has exactly six terms (see the Monthly reference). phi(n)=A000010(n); c(n)=A007947(n).

Examples

			8 is in the sequence because phi(8)=4 (1,3,5,7), c(8)=2 (2 being the only prime divisor of 8) and so phi(8)=c(8)^2.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 108, p. 38, Ellipses, Paris 2008.
• J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 745 ; pp 95; 317-8, Ellipses Paris 2004.
• J.-M. De Koninck & A. Mercier, 1001 Problems in Classical Number Theory, Problem 745 ; pp 80; 273-4, Amer. Math. Soc. Providence RI 2007.

Links

• J.-M. De Koninck, When the Totient Is the Product of the Squared Prime Divisors: Problem 10966, Amer. Math. Monthly, 111 (2004), p. 536.

Crossrefs

Cf. A000010, A007947.

Programs

Maple

with(numtheory): c:=proc(n) local div: div:=convert(factorset(n),list): product(div[j],j=1..nops(div)) end:p:=proc(n) if phi(n)=c(n)^2 then n else fi end: seq(p(n),n=1..42000);

Mathematica

Select[Range[42000], EulerPhi[#] == Times @@ FactorInteger[#][[All,1]]^2 & ] (* Jean-François Alcover, Sep 12 2011 *)