A306616 Integers k such that phi(Catalan(k+1)) = 4*phi(Catalan(k)) where phi is A000010 and Catalan is A000108.

2, 8, 19, 20, 36, 42, 44, 55, 56, 76, 91, 109, 116, 120, 140, 143, 152, 156, 176, 184, 200, 204, 213, 216, 224, 235, 242, 260, 289, 296, 300, 380, 384, 400, 401, 415, 436, 464, 469, 476, 524, 547, 553, 564, 595, 602, 616, 624, 630, 631, 660, 685, 704, 716, 744, 776, 800

Offset: 1

Author

Michel Marcus, Mar 01 2019

Keywords

• nonn

Comments

Integers k such that A062624(k+1) = 4*A062624(k).

Consists of integers k (see p. 1405 of Luca link):

k = 2p-2, where p >= 5 is a prime such that q = 4p-3 is also prime (see A157978);

k = 3p-2, where p > 5 is a prime such that q = 2p-1 is also prime (see A005382).

Examples

			phi(C(2)) = phi(2) = 1 and phi(C(3)) = phi(5) = 4 so 2 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Florian Luca and Pantelimon Stanica, On the Euler function of the Catalan numbers, Journal of Number Theory, Vol. 132, No. 7 (2012), pp. 1404-1424.

Crossrefs

Cf. A000010, A000108, A062624.

Cf. A005382, A157978.

Programs

Mathematica

Select[Range[1000], EulerPhi[CatalanNumber[#+1]]== 4*EulerPhi[CatalanNumber[#]] &] (* G. C. Greubel, Mar 02 2019 *)

PARI

C(n) = binomial(2*n,n)/(n+1);
isok(n) = eulerphi(C(n+1)) == 4*eulerphi(C(n));

Sage

[n for n in (1..1000) if euler_phi(catalan_number(n+1)) == 4*euler_phi(catalan_number(n))] # G. C. Greubel, Mar 02 2019