A369171 Numbers k such that A000688(k) = A046660(k+1).

3, 11, 17, 19, 43, 51, 59, 62, 67, 74, 83, 89, 91, 97, 99, 115, 123, 124, 131, 139, 146, 149, 155, 163, 170, 174, 187, 188, 197, 203, 206, 211, 219, 227, 233, 235, 241, 259, 267, 274, 278, 279, 283, 291, 293, 305, 307, 314, 331, 337, 339, 341, 342, 347, 349, 350

Offset: 1

Author

Amiram Eldar, Jan 15 2024

Keywords

• nonn
• easy

Comments

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... .

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, 51, Number Theory, Budapest, 1987, pp. 45-91. See p. 60.

Crossrefs

Cf. A000688, A046660.

Programs

Mathematica

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

PARI

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

Formula

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).