A191278 Count of mosaic numbers that equal n.

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 6, 1, 3, 3, 1, 1, 6, 1, 6, 3, 3, 1, 10, 1, 3, 1, 6, 1, 16, 1, 1, 3, 3, 3, 20, 1, 3, 3, 10, 1, 16, 1, 6, 6, 3, 1, 15, 1, 6, 3, 6, 1, 10, 3, 10, 3, 3, 1, 50, 1, 3, 6, 1, 3, 16, 1, 6, 3, 16, 1, 50, 1, 3, 6, 6, 3, 16, 1, 15, 1, 3, 1, 50, 3, 3, 3, 10, 1, 50

Offset

1,6

Comments

The number of solutions x to A000026(x) = n.

a(n) depends only on the prime signature of n (A118914). - Amiram Eldar, Dec 02 2025

Links

R. J. Mathar, Table of n, a(n) for n = 1..1000

B. Gordon and M. M. Robertson, Two theorems on mosaics, Canad. J. Math., Vol. 17 (1965), pp. 1010-1014.

Formula

Let n = Product_j p_j^e(j) be the prime factorization of n and beta(n) = A073093(n). Then a(n) = (Product_j binomial(beta,e(j))) / beta(n). [Gordon and Robertson, 1965, Theorem 1]

Maple

A191278 := proc(n)
    local f, beta, a, j ;
    f := ifactors(n)[2] ;
    beta := A073093(n) ;
    a := 1/beta ;
    for j in ifactors(n)[2] do
        a := a*binomial(beta, op(2, j) ) ;
    end do:
    a ;
end proc:

Mathematica

a[n_] := Module[{e = FactorInteger[n][[;; , 2]], beta}, beta = Total[e] + 1; Times @@ Binomial[beta, e]/beta]; Array[a, 100] (* Amiram Eldar, Dec 02 2025 *)

Prog

(PARI) a(n) = {my(e = factor(n)[, 2], beta = vecsum(e) + 1); prod(i = 1, #e, binomial(beta, e[i])) / beta; } \\ Amiram Eldar, Dec 02 2025

Crossrefs

Cf. A000026, A073093, A118914.

Sequence in context: A291448 A114006 A050328 * A363085 A359577 A030401

Adjacent sequences: A191275 A191276 A191277 * A191279 A191280 A191281

Keyword

nonn,easy,changed

Author

R. J. Mathar, May 29 2011

Status

approved