A369171 - OEIS

%I #7 Jan 15 2024 09:42:33

%S 3,11,17,19,43,51,59,62,67,74,83,89,91,97,99,115,123,124,131,139,146,

%T 149,155,163,170,174,187,188,197,203,206,211,219,227,233,235,241,259,

%U 267,274,278,279,283,291,293,305,307,314,331,337,339,341,342,347,349,350

%N Numbers k such that A000688(k) = A046660(k+1).

%C The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 15, 150, 1548, 15499, 154916, 1549105, 15489932, 154901767, 1549014294, ... . From these values the asymptotic density of this sequence, whose existence was proven by Erdős and Ivić (1987) (the constant c in the Formula section), can be empirically evaluated by 0.15490... .

%D József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter XIII, p. 476.

%H Amiram Eldar, <a href="/A369171/b369171.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Erdős and Aleksandar Ivić, <a href="http://combinatorica.hu/~p_erdos/1987-32b.pdf">The distribution of values of a certain class of arithmetic functions at consecutive integers</a>, Colloq. Math. Soc. János Bolyai, 51, Number Theory, Budapest, 1987, pp. 45-91. See p. 60.

%F The number of terms not exceeding x, N(x) = c * x + O(x^(3/4) * log(x)^4), where c > 0 is a constant (Erdős and Ivić, 1987).

%t Select[Range[350], FiniteAbelianGroupCount[#] == PrimeOmega[#+1] - PrimeNu[#+1] &]

%o (PARI) is(n) = vecprod(apply(numbpart, factor(n)[, 2])) == bigomega(n+1) - omega(n+1);

%Y Cf. A000688, A046660.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Jan 15 2024