A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

2, 39, 1279786

Offset

1,1

Links

Table of n, a(n) for n=1..3.

Example

The first two squarefree numbers are 1 and 2; 1 has 0 prime factors and 2 has 1 prime factor, so a(1)=2.
At k=39, in the interval [1..k], there are 12 squarefree numbers with 1 prime factor (i.e., 12 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37), and 12 squarefree numbers with 2 prime factors (i.e., 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39). k=39 is the smallest such positive number for which these two counts are the same (and are positive), so a(2)=39.
At k=1279786, the interval [1..k] includes 265549 squarefree numbers with 2 prime factors and the same number of squarefree numbers with 3 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(3)=1279786.

Prog

(PARI) a(n) = my(nbm = 0, nbn = 0); for (k=1, oo, if (issquarefree(k), my(o=omega(k)); if (o==n, nbn++); if (o==n-1, nbm++); if (nbm && (nbn==nbm), return(k)))); \\ Michel Marcus, Nov 25 2022

Crossrefs

Cf. A005117, A072047.

Cf. 1 to 5 distinct primes: A000040, A006881, A007304, A046386, A046387.

Cf. 6 to 10 distinct primes: A067885, A123321, A123322, A115343, A281222.

Cf. A340316.

Sequence in context: A247878 A339774 A232086 * A047660 A213909 A349717

Adjacent sequences: A356680 A356681 A356682 * A356684 A356685 A356686

Keyword

nonn,bref,hard,more

Author

Jon E. Schoenfield, Nov 22 2022

Status

approved