A101638 Number of distinct 4-almost primes A014613 which are factors of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1

Offset

1,48

Comments

This is the inverse Moebius transform of A101637. If we take the prime factorization of n = (p1^e1)*(p2^e2)* ... * (pj^ej) then a(n) = |{k: ek>=4}| + ((j-1)/2)*|{k: ek>=3}| + C(|{k: ek>=2}|,2) + C(j,4). The first term is the number of distinct 4th powers of primes in the factors of n (the first way of finding a 4-almost prime). The second term is the number of distinct cubes of primes, each of which can be multiplied by any of the other distinct primes, halved to avoid double-counts (the second way of finding a 4-almost prime). The third term is the number of distinct pairs of squares of primes in the factors of n (the third way of finding a 4-almost prime). The 4th term is the number of distinct products of 4 distinct primes, which is the number of combinations of j primes in the factors of n taken 4 at a time, A000332(j), (the 4th way of finding a 4-almost prime).

References

Hardy, G. H. and Wright, E. M. Section 17.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

E. A. Bender and J. R. Goldman, On the Applications of Moebius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, 789-803, 1975.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Eric Weisstein's World of Mathematics, Almost Prime.

Eric Weisstein's World of Mathematics, Moebius Transform..

Example

a(96) = 2 because 96 = 16 * 6 hence divisible by the 4-almost prime 16 and also 96 = 24 * 4 hence divisible by the 4-almost prime 24.

Prog

(PARI) a(n)=my(f=factor(n)[, 2], v=apply(k->sum(i=1, #f, f[i]>k), [0..3])); v[4] + v[3]*(v[1]-1) + binomial(v[2], 2) + v[2]*binomial(v[1]-1, 2) + binomial(v[1], 4) \\ Charles R Greathouse IV, Sep 14 2015

Crossrefs

Cf. A101638, A014613, A000332, A086971, A100605, A000040, A001358, A014612, A014614.

Sequence in context: A277148 A194024 A082786 * A321379 A070141 A088722

Adjacent sequences: A101635 A101636 A101637 * A101639 A101640 A101641

Keyword

easy,nonn

Author

Jonathan Vos Post, Dec 10 2004

Status

approved