A216153 The partial products of a(n) are the distinct values of the exponential of the von Mangoldt function modified by restricting the divisors to prime divisors (A205957).

1, 2, 6, 4, 3, 10, 24, 14, 15, 8, 54, 40, 21, 22, 96, 5, 26, 9, 56, 900, 16, 33, 34, 35, 216, 38, 39, 160, 1764, 88, 135, 46, 384, 7, 250, 51, 104, 486, 55, 224, 57, 58, 7200, 62, 189, 32, 65, 4356, 136, 69, 4900, 864, 74, 375, 152, 77, 6084, 640, 27, 82

Offset

1,2

Comments

The partial products of a(n) are A216152(n) which are the distinct values of the 'prime lcm(n)' A205957.

Let b(n) denote the nonprime numbers A018252(n).

If n = 1 then a(n) = b(n) = 1

else if a(n) < b(n) then

a(n) is a cototient of consecutive pure powers of primes (A053211),

b(n) is a prime power with exponent > 1 (A025475),

b(n)/a(n) is a prime root of n-th nontrivial prime power (A025476);

else if a(n) > b(n) then

b(n) is a number which is neither a prime power nor a semiprime (A102467);

else if a(n) = b(n) then

a(n) is the product of two distinct primes (A006881).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Peter Luschny, The von Mangoldt Transformation.

Formula

a(n) = A205957(A018252(n))/A205957(A018252(n-1)) for n > 1, a(1) = 1.

Mathematica

A205957[n_] := Exp[-Sum[ MoebiusMu[p]*Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]]; nonPrime[1] = 1; nonPrime[n_] := Which[k0 = k /. FindRoot[ n + PrimePi[k] == k , {k, n}] // Floor; n+PrimePi[k0] == k0, k0 , n+PrimePi[k0+1] == k0+1, k0+1, n+PrimePi[k0+2] == k0+2, k0+2, True, k0]; a[1] = 1; a[n_] := A205957[nonPrime[n]] / A205957[nonPrime[n-1]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jun 27 2013 *)

Prog

(Sage)
def A216153(n):
    if n == 1 : return 1
    return A205957(A018252(n))/A205957(A018252(n-1))

Crossrefs

Cf. A205957, A205959, A216152.

Sequence in context: A102510 A206225 A208507 * A100115 A029670 A160205

Adjacent sequences: A216150 A216151 A216152 * A216154 A216155 A216156

Keyword

nonn,easy

Author

Peter Luschny, Sep 02 2012

Status

approved