A062624 -id:A062624 - cp's OEIS frontend

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

Michel Marcus, Mar 01 2019

nonn

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).

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

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.

Cf. A000010, A000108, A062624.

Cf. A005382, A157978.

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

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

[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

cp's OEIS Frontend

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

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