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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). 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified August 4 00:29 EDT 2021. Contains 346441 sequences. (Running on oeis4.)