cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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).

Original entry on oeis.org

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

Views

Author

Peter Luschny, Sep 02 2012

Keywords

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

Crossrefs

Cf. A205957, A205959, A216152.

Programs

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

Formula

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

A216152 A205957(n) where n is a nonprime number.

Original entry on oeis.org

1, 2, 12, 48, 144, 1440, 34560, 483840, 7257600, 58060800, 3135283200, 125411328000, 2633637888000, 57940033536000, 5562243219456000, 27811216097280000, 723091618529280000, 6507824566763520000, 364438175738757120000, 327994358164881408000000
Offset: 1

Views

Author

Peter Luschny, Sep 02 2012

Keywords

Comments

The distinct values of A205957. Partial products of A216153.
a(1),...,a(10) are highly totient numbers (A097942) and products of distinct factorials (A058295). The author conjectures that this is true in general.

Links

Crossrefs

Cf. A051451.

Programs

  • Mathematica
    A205957[n_] := Exp[-Sum[MoebiusMu[p] Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]];
Table[A205957[n], {n, 0, 30}] // DeleteDuplicates (* Jean-François Alcover, Jul 08 2019 *)
  • Sage
    # sorted(list(set([A205957(n) for n in (0..31)])))
def A216152_list(n) :
    C = filter(lambda k: not is_prime(k), (1..n))
    return [A205957(c) for c in C]
A216152_list(31)

Formula

a(n) = A205957(A018252(n)).

A025527 a(n) = n!/lcm{1,2,...,n} = (n-1)!/lcm{C(n-1,0), C(n-1,1), ..., C(n-1,n-1)}.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 12, 48, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 522547200, 522547200, 10450944000, 219469824000, 4828336128000, 4828336128000, 115880067072000, 579400335360000, 15064408719360000
Offset: 1

Views

Author

Clark Kimberling, Dec 11 1999

Keywords

Comments

a(n) = a(n-1) iff n is prime. Thus a(1)=a(2)=a(3)=1 is the only triple in this sequence. - Franz Vrabec, Sep 10 2005
a(k) = a(k+1) for k in A006093. - Lekraj Beedassy, Aug 03 2006
Partial products of A048671. - Peter Luschny, Sep 09 2009

Examples

			a(5) = 2 as 5!/lcm(1..5) = 120/60 = 2.

Links

Crossrefs

Cf. A000142, A002541, A006093, A003418, A048671, A025529, A205957.
See also A002944, A025537.

Programs

Formula

a(n) = A000142(n)/A003418(n) = A000254(n)/A025529(n). - Franz Vrabec, Sep 13 2005
log a(n) = n log n - 2n + O(n/log^4 n). (The error term can be improved. On the Riemann Hypothesis it is O(n^k) for any k > 1/2.) - Charles R Greathouse IV, Oct 16 2012
a(n) = A205957(n), 1 <= n <= 11. - Daniel Forgues, Apr 22 2014
Conjecture: a(A006093(n)) = phi(A000142(A006093(n))) / phi(A003418(A006093(n))), where phi is the Euler totient function. - Fred Daniel Kline, Jun 03 2017

A205959 a(n) = n^omega(n)/rad(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 4, 3, 10, 1, 24, 1, 14, 15, 8, 1, 54, 1, 40, 21, 22, 1, 96, 5, 26, 9, 56, 1, 900, 1, 16, 33, 34, 35, 216, 1, 38, 39, 160, 1, 1764, 1, 88, 135, 46, 1, 384, 7, 250, 51, 104, 1, 486, 55, 224, 57, 58, 1, 7200, 1, 62, 189, 32, 65, 4356, 1, 136
Offset: 1

Views

Author

Peter Luschny, Feb 03 2012

Keywords

Comments

a(n) = exp(-Sum_{d in P} moebius(d)*log(n/d)) where P = {d : d divides n and d is prime}. This is a variant of the (exponential of the) von Mangoldt function where the divisors are restricted to prime divisors. The (exponential of the) summatory function is A205957. Apart from n=1 the value is 1 if and only if n is prime; the fixed points are the products of two distinct primes (A006881).

Links

Crossrefs

Cf. A003418, A025527, A008578, A102467, A006881, A205957.

Programs

  • Haskell
    a205959 n = product $ map (div n) $ a027748_row n
-- Reinhard Zumkeller, Dec 15 2013
  • Maple
    with(numtheory): A205959 := proc(n) select(isprime, divisors(n));
simplify(exp(-add(mobius(d)*log(n/d), d=%))) end:
# Alternative:
a := n -> local p; mul(n/p[1], p in ifactors(n)[2]):
seq(a(n), n = 1..68); # Peter Luschny, Jul 19 2023
  • Mathematica
    a[n_] := Exp[-Sum[ MoebiusMu[d]*Log[n/d], {d, FactorInteger[n][[All, 1]]}]]; Table[a[n], {n, 1, 68}] (* Jean-François Alcover, Jan 15 2013 *)
  • PARI
    a(n)=my(f=factor(n)[,1]);prod(i=1,#f,n/f[i]) \\ Charles R Greathouse IV, Jun 27 2013
  • Python
    from math import prod
from sympy import primefactors
def A205959(n): return prod(n//p for p in primefactors(n)) # Chai Wah Wu, Jul 12 2023
  • Sage
    def A205959(n) :
    P = filter(is_prime, divisors(n))
    return simplify(exp(-add(moebius(d)*log(n/d) for d in P)))
[A205959(n) for n in (1..60)]

Formula

a(n) = Product_{p|n} n/p. - Charles R Greathouse IV, Jun 27 2013
a(n) = Product_{k=1..A001221(n)} n/A027748(n,k). - Reinhard Zumkeller, Dec 15 2013
If n is squarefree, then a(n) = n^(omega(n)-1). - Wesley Ivan Hurt, Jun 09 2020
a(p^e) = p^(e-1) for p prime, e > 0. - Bernard Schott, Jun 09 2020

Extensions

New name from Charles R Greathouse IV, Jun 30 2013
Showing 1-4 of 4 results.

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