A172314 Numbers k such that phi(k+1) = 4*phi(k).

1260, 13650, 17556, 18720, 24510, 42120, 113610, 244530, 266070, 712080, 749910, 795690, 992250, 1080720, 1286730, 1458270, 1849470, 2271060, 2457690, 3295380, 3370770, 3414840, 3714750, 4061970, 4736490, 5314050, 5827080, 6566910, 6935082, 7303980, 7864080

Offset: 1

Author

Michel Lagneau, Jan 31 2010

Keywords

• nonn

Examples

			phi(1260) = 288. phi(1261) = 1152. 4*phi(1260) = phi(1261).
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.
• R. K. Guy, Unsolved Problems Number Theory, Sect. B36.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..300
• V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), 332.
• M. Lal and P. Gillard, On the equation phi(n) = phi(n+k), Math. Comp. 26 (1972), 579-583.
• K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. De Pauw University, 1972. [ Cf. Review on Math. Comp., Vol. 27, p. 447, 1973 ].
• L. Moser, Some equations involving Euler's totient function, Amer. Math. Monthly, 56 (1949), 22-23.
• A. Shinzel, Sur l'équation phi(x+k) = phi(x), Acta Arith. 4 (1958), 181-184, [MR0106857]

Crossrefs

Cf. A001274, A050472, A067143.

Programs

Magma

[n: n in [1..2*10^6] | EulerPhi(n+1) eq 4*EulerPhi(n)]; // Vincenzo Librandi, Jan 27 2016

Maple

with(numtheory): for n from 1 to 4000000 do; if 4*phi(n) = phi(n+1) then print(n); else fi ; od;

Mathematica

#[[1,1]]&/@Select[Partition[Table[{n,EulerPhi[n]},{n,4000000}],2,1], 4#[[1,2]]==#[[2,2]]&] (* Harvey P. Dale, Oct 11 2011 *)
Select[Range@1000000, EulerPhi@# 4 == EulerPhi[# + 1] &] (* Vincenzo Librandi, Jan 27 2016 *)

Extensions

References separated by R. J. Mathar, Feb 19 2010