A308378 Numbers k such that phi(2k+1) = phi(2k+2).

0, 1, 7, 127, 247, 487, 1312, 1627, 1852, 2593, 5857, 6682, 9157, 11467, 12772, 23107, 24607, 24667, 28822, 32767, 82087, 92317, 99157, 107887, 143497, 153697, 159637, 194122, 198742, 207637, 245767, 284407, 294703, 343492, 420127

Offset: 1

Author

Torlach Rush, May 24 2019

Keywords

• nonn

Comments

For n > 0, 2*a(n) + 1 is a term of A020884. This is because 2*a(n) + 1 is odd and every odd number is the difference of the squares of two consecutive numbers and hence are coprime.

For n > 0, (2*a(n) + 1) * (2*a(n) + 2) is a term of A024364. This is because (2*a(n) + 1) * (2*a(n) + 2) = 2*((a(n) + 1)^2 + (a(n) + 1) * a(n)) and gcd((a(n) + 1), a(n)) = 1.

For n > 0, a(n) is congruent to 1 or 4 mod 6.

2*a(n) + 1 is congruent to 1 or 3 mod 6 and is a term of A047241.

2*a(n) + 2 is congruent to 2 or 4 mod 6 and is a term of A047235.

Examples

			0 is a term because phi(1) = phi(2) = 1.
1 is a term because phi(3) = phi(4) = 2.
7 is a term because phi(15) = phi(16) = 8.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5416 (calculated using the b-file at A001274)

Crossrefs

Cf. A000010, A004767, A020884, A024364, A047235, A047241, A299535.

Subset of A047234.

Subset of A001274.

Programs

Mathematica

Select[Range[0, 9999], EulerPhi[2# + 1] == EulerPhi[2# + 2] &] (* Alonso del Arte, Jul 05 2019 *)
Select[(#-1)/2&/@SequencePosition[EulerPhi[Range[900000]],{x_,x_}][[All,1]],IntegerQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 24 2019 *)

PARI

lista(nn) = for(n=0, nn, if(eulerphi(2*n+1) == eulerphi(2*n+2), print1(n, ", ")));
lista(430000)

Formula

a(n) = (A299535(n) - 2) / 2.