A330292 a(n) = number of integers 1 <= k < n such that omega(k) <= omega(n), where omega = A001221.

0, 1, 2, 3, 4, 5, 5, 6, 7, 9, 8, 11, 9, 13, 14, 10, 11, 17, 12, 19, 20, 21, 13, 23, 14, 25, 15, 27, 16, 29, 17, 18, 31, 32, 33, 34, 19, 36, 37, 38, 20, 41, 21, 41, 42, 43, 22, 45, 23, 47, 48, 49, 24, 51, 52, 53, 54, 55, 25, 59, 26, 58, 59, 27, 61, 65, 28, 63, 64

Offset

1,3

Comments

Any natural number n can be represented as n = (k_1)^p_1 * (k_2)^p_2 * ... * (k_h)^p_h, where k_i is prime for any i from 1 to h. Let us consider the function omega(n) = h, which represents the number of distinct prime factors of n. Then a(k) is the number of positive integers j less than k for which the value of function omega(j) is <= omega(k).

a(P) = A025528(P) for P a prime power in A246655.

a(Q) = Q - 1 for Q a primorial number in A002110.

Let us consider n > k such that omega(n) = omega(k) = omega and there is no w such that n > w > k and omega(w) > omega. Hence a(n) - a(k) = n - k.

Links

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Example

a(1) = 0: 1 has no predecessor, omega(1) = 0 by convention;
a(2) = 1 because omega(2) = 1, 1 >= omega(0);
a(3) = 2 because omega(3) = 1 and none of omega(1), omega(2) >= 1;
a(4) = 3 because omega(4) = 1 and none of omega(1), omega(2), omega(3) >= 1.

Mathematica

a[n_] := Block[{t = PrimeNu[n]}, Length@ Select[Range[n - 1], PrimeNu[#] <= t &]]; Array[a, 70] (* Giovanni Resta, Dec 10 2019 *)

Prog

(Python)
def primes(n):
    divisors = [ d for d in range(2, n//2+1) if n % d == 0 ]
    return [ d for d in divisors if \
             all( d % od != 0 for od in divisors if od != d ) ]
pprimes = {}
for i in range(1, 10000):
    res = len(primes(i))
    if res == 0:
        res = 1
    pprimes[i] = res
for k in range(1, 10000):
    s = 0
    for i in range(1, k):
        if pprimes[i] <= pprimes[k]:
            s+=1
    print(s)
(PARI) for(n=1, 70, my(omn=omega(n), m=0); for(k=1, n-1, if(omega(k)<=omn, m++)); print1(m, ", ")) \\ Hugo Pfoertner, Dec 10 2019
(PARI) a(n) = my(omn=omega(n)); sum(k=1, n-1, omega(k) <= omn); \\ Michel Marcus, Dec 11 2019

Crossrefs

Cf. A001221 (omega), A025528, A246655, A002110.

Sequence in context: A291811 A291812 A363694 * A017866 A086886 A017840

Adjacent sequences: A330289 A330290 A330291 * A330293 A330294 A330295

Keyword

nonn,easy

Author

Dilshod Urazov, Dec 10 2019

Extensions

More terms from Giovanni Resta, Dec 10 2019

Status

approved