A292208 Composite numbers k such that sigma(cototient(k)) = cototient(sigma(k) - k) + cototient(k); that is, f(g(k)) = g(f(k)) where f = A001065 and g = A051953.

4, 16, 35, 65, 77, 78, 114, 146, 161, 185, 209, 221, 256, 335, 341, 371, 377, 437, 485, 515, 595, 611, 626, 644, 654, 671, 707, 731, 767, 779, 805, 851, 899, 917, 965, 1007, 1067, 1115, 1157, 1211, 1247, 1271, 1309, 1337, 1385, 1397, 1463, 1495, 1529, 1535, 1577, 1631, 1645, 1691, 1771

Offset: 1

Author

Altug Alkan, Sep 11 2017

Keywords

• nonn

Comments

Luca and Pomerance proved that arithmetic functions f(g(n)) and g(f(n)) are independent where f = A001065 and g = A051953. For related details and theorems see Luca & Pomerance link.

Examples

			35 = 5*7 is a term because A001065(A051953(35)) = A051953(A001065(35)).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Florian Luca and Carl Pomerance, Local behavior of the composition of the aliquot and co-totient functions, in: G. Andrews and F. Garvan (eds.), Analytic Number Theory, Modular Forms and q-Hypergeometric Series, ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham, 2017; author's copy.

Crossrefs

Cf. A000203, A001065, A033632, A051953.

Programs

Mathematica

Select[Range@ 1800, Function[n, And[CompositeQ@ n, DivisorSigma[1, n - EulerPhi@ n] == (n - EulerPhi@ n) + # - EulerPhi@ # &[DivisorSigma[1, n] - n]]]] (* Michael De Vlieger, Sep 12 2017 *)

PARI

a001065(n) = sigma(n)-n;
a051953(n) = n-eulerphi(n);
lista(nn) = forcomposite(n=4, nn, if(a051953(a001065(n))==a001065(a051953(n)), print1(n, ", ")));