OFFSET
1,5
COMMENTS
The second smallest prime divisor of a number k is the second member in the ordered list of the distinct prime divisors of k. All the numbers that are not prime powers (A000961) have a second smallest prime divisor.
REFERENCES
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 337-341.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..376
Paul Erdős and Gérald Tenenbaum, Sur les densités de certaines suites d'entiers, Proc. London Math. Soc. (3), Vol. 59, No. 3 (1989), pp. 417-438; alternative link.
FORMULA
EXAMPLE
The fractions begin with 0, 1/6, 1/10, 1/15, 46/1155, 44/1365, 288/12155, 33216/1616615, 613248/37182145, 151296/11849255, 391584768/33426748355, ...
a(1) = 0 since there are no numbers whose second smallest prime divisor is prime(1) = 2.
a(2)/A342480(2) = 1/6 since the numbers whose second smallest prime divisor is prime(2) = 3 are the positive multiples of 6.
a(3)/A342480(3) = 1/10 since the numbers whose second smallest prime divisor is prime(3) = 5 are the numbers congruent to {10, 15, 20} (mod 30) whose density is 3/30 = 1/10.
MATHEMATICA
f[n_] := Module[{p = Prime[n], q}, q = Select[Range[p - 1], PrimeQ]; Plus @@ (1/(q - 1))*Times @@ ((q - 1)/q)/p]; Numerator @ Array[f, 30]
KEYWORD
nonn,easy,frac
AUTHOR
Amiram Eldar, Mar 13 2021
STATUS
approved