A205959 -id:A205959 - cp's OEIS frontend

A205957 a(n) = exp(-Sum_{k=1..n} Sum_{d|k, d prime} moebius(d)*log(k/d)).

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

Offset: 0

Peter Luschny, Sep 01 2012

nonn

The author proposes to denote this sequence lcm_{p}(n) as lcm(n) = lcm({1,2,..n}) = exp(Sum_{k=1..n} Sum_{d|k} moebius(d)*log(k/d)).

For n > 0 the a(n) are the partial products of A205959(n), which is the exponential of a modified von Mangoldt function where the divisors are restricted to prime divisors.

Michel Marcus, Table of n, a(n) for n = 0..400
Peter Luschny, The von Mangoldt Transformation.

Cf. A003418, A205959, A216152, A216153.

with(numtheory):
A205957 := proc(n) simplify(exp(-add(add(mobius(d)*log(k/d), d=select(isprime, divisors(k))),k=1..n))) end: seq(A205957(i), i=0..27);

a[n_] := Exp[-Sum[ MoebiusMu[p] Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 27 2013 *)

a(n)=prod(k=4,n,my(f=factor(k)[, 1]); prod(i=1, #f, k/f[i])) \\ Charles R Greathouse IV, Jun 27 2013

def A205957(n) : return simplify(exp(-add(add(moebius(p)*log(k/p) for p in prime_divisors(k)) for k in (1..n))))

a(n) = Product_{p prime, p<=n} (floor(n/p)!). - Ridouane Oudra, Nov 22 2021

A363923 a(n) = n^npf(n) / rad(n), where npf(n) is the number of prime factors with multiplicity of n.

Original entry on oeis.org

1, 1, 1, 8, 1, 6, 1, 256, 27, 10, 1, 288, 1, 14, 15, 32768, 1, 972, 1, 800, 21, 22, 1, 55296, 125, 26, 6561, 1568, 1, 900, 1, 16777216, 33, 34, 35, 279936, 1, 38, 39, 256000, 1, 1764, 1, 3872, 6075, 46, 1, 42467328, 343, 12500, 51, 5408, 1, 1417176, 55, 702464

Offset: 1

Peter Luschny, Jul 11 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001222, A007947, A205959, A008578.

with(NumberTheory): a := n -> n^NumberOfPrimeFactors(n) / Radical(n):
seq(a(n), n = 1..56);

Array[#^PrimeOmega[#]/(Times @@ FactorInteger[#][[All, 1]]) &, 56] (* Michael De Vlieger, Jul 11 2023 *)

a(n) = my(f=factor(n)); n^bigomega(f)/factorback(f[, 1]); \\ Michel Marcus, Jul 11 2023

from math import prod
from sympy import factorint
def A363923(n): return prod(n**e//p for p, e in factorint(n).items()) # Chai Wah Wu, Jul 12 2023

a(n) = n^A001222(n) / A007947(n).

a(n) = 1 <=> n term of A008578.

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

Peter Luschny, Sep 02 2012

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Peter Luschny, The von Mangoldt Transformation.

Cf. A205957, A205959, A216152.

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 *)

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

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

A363919 a(n) = n^excess(n), where excess(n) = A046660(n).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 64, 9, 1, 1, 12, 1, 1, 1, 4096, 1, 18, 1, 20, 1, 1, 1, 576, 25, 1, 729, 28, 1, 1, 1, 1048576, 1, 1, 1, 1296, 1, 1, 1, 1600, 1, 1, 1, 44, 45, 1, 1, 110592, 49, 50, 1, 52, 1, 2916, 1, 3136, 1, 1, 1, 60, 1, 1, 63, 1073741824, 1, 1, 1, 68

Offset: 1

Peter Luschny and Michael De Vlieger, Jul 14 2023

nonn

			108 = 2^2 * 3^3 => excess(108) = 5 - 2 => a(108) = 108^3 = 1259712.

Plot2 A363923 vs A205959.
Michael De Vlieger, Plot p(k)^e(k) at (x,y) = (n,k) for n = 1..2^11, with a color function representing e(k) where black represents e(k) = 1, red e(k) = 2, through magenta. The bar at bottom represents a(n) = 1 in black, a(n) a composite prime power in gold, a(n) neither squarefree nor semiprime in blue, and a(n) such that all e(k) > 1 in purple.
Michael De Vlieger, Notes on certain sequences relating BigOmega(n), omega(n), and rad(n), 2023.

Cf. A001221, A001222, A005117, A046660, A053763, A056170, A060687, A013929, A205959, A337945, A363923.

using Nemo
exc(n::fmpz) = sum(e - 1 for (p, e) in factor(n))
A363919(n::fmpz) = n < 2 ? fmpz(1) : n^exc(n)
println([A363919(fmpz(n)) for n in 1:68])

with(NumberTheory):
A363919 := n -> n^(NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct')):
# Alternative:
a := n -> local i: n^add(i[2] - 1, i in ifactors(n)[2]): seq(a(n), n = 1..68);

Array[#^(PrimeOmega[#] - PrimeNu[#]) &, 120]

a(n) = my(f=factor(n)[, 2]); n^(vecsum(f)-#f); \\ Michel Marcus, Jul 16 2023

from sympy import factorint
def A363919(n): return n**sum(map(lambda e:e-1,factorint(n).values())) # Chai Wah Wu, Jul 18 2023

def A363919(n):
    if n < 2: return 1
    return n^sum(p[1] - 1 for p in list(factor(n)))
print([A363919(n) for n in srange(1, 69)])

a(n) = n^(Sum_{p in Factors(n)} (mult(p) - 1)), where Factors(n) is the integer factorization of n and mult(p) the multiplicity of the prime factor p.
a(n) = A363923(n) / A205959(n).
a(n) = n^A046660(n) = n^(A001222(n) - A001221(n)).
a(n) = 1 or divisible by at least one squared prime.
a(n) = 1 <=> n is squarefree (A005117).
a(n) != 1 <=> A056170(n) != 0.
a(n) = n <=> n = A060687(n - 1) for n >= 2.
a(2^n) = 2^(n*(n - 1)) = A053763(n).
a(n) <= 2^(lb(n)*(lb(n)-1)), where lb(n) = floor(log_{2}(n)).
a(n) is even <=> n = 2*A337945(n).
a(n) > 1 is odd <=> n = A053850(n).
n is prime => a(n) = 1. ('prime' means term of A000040).
n is prime product => a(n) = 1. ('prime product' means term of A144338).
n is proper prime power => a(n) is proper prime power. ('proper prime power' means term of A246547).
Moebius(a(n)) = [a(n) = 1], where [ ] denotes the Iverson bracket.

A205958 a(0) = 1 and a(n) = A180000(n)*a(floor(n/2))^2 for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 48, 16, 40, 40, 270, 270, 945, 63, 129024, 129024, 64512, 64512, 2016000, 96000, 528000, 528000, 144342000, 28868400, 187644600, 20849400, 1787836050, 1787836050, 59594535, 59594535, 3999321544458240, 121191561953280, 1030128276602880

Offset: 0

Peter Luschny, Feb 04 2012

nonn

lcm(1,2,..,n) = (n!*a(n)) / ((n/2)!*a(n/2))^2.

lcm(1,2,..,n)*a(n) is a divisor of n! and n!/(lcm(1,2,..,n)*a(n)) is a square.

Cf. A180000, A025527, A205959.

def A205958(n) :
    if n == 0 : return 1
    return A180000(n)*A205958(n//2)^2

A363917 a(n) = Product_{p in Factors(n)} mult(p) * n^mult(p) / p, where Factors(n) is the integer factorization of n and mult(p) the multiplicity of the prime factor p.

Original entry on oeis.org

1, 1, 1, 16, 1, 6, 1, 768, 54, 10, 1, 576, 1, 14, 15, 131072, 1, 1944, 1, 1600, 21, 22, 1, 165888, 250, 26, 19683, 3136, 1, 900, 1, 83886080, 33, 34, 35, 1119744, 1, 38, 39, 768000, 1, 1764, 1, 7744, 12150, 46, 1, 169869312, 686, 25000, 51, 10816, 1, 4251528

Offset: 1

Peter Luschny, Jul 19 2023

nonn

Cf. A363923, A005361, A363918, A205959.

A363917 := n-> local p; mul(p[2] * n^p[2] / p[1], p in ifactors(n)[2]):
seq(A363917(n), n = 1..54);

a(n) = A363918(n) * A205959(n).

a(n) = A363923(n) * A005361(n).

A382941 a(n) = exp(Sum_{d|n} A382883(d)*log(n/d)).

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 4, 3, 1, 11, 3, 13, 1, 1, 16, 17, 2, 19, 5, 1, 1, 23, 18, 5, 1, 9, 7, 29, 1, 31, 64, 1, 1, 1, 36, 37, 1, 1, 50, 41, 1, 43, 11, 5, 1, 47, 72, 7, 2, 1, 13, 53, 12, 1, 98, 1, 1, 59, 15, 61, 1, 7, 512, 1, 1, 67, 17, 1, 1, 71, 648, 73, 1, 3, 19

Offset: 1

Peter Luschny, Apr 09 2025

nonn

See the comments in A382883.

Cf. A000469, A014963, A205959, A382883, A383263.

h := proc(n) option remember; local j; ifelse(n = 1, 1,
-add(ifelse(j = 1, 1, padic:-ordp(n, j))*h(j), j = 1..n-1)) end:
a := n -> local d; simplify(exp(add(h(d)*log(n/d), d in numtheory:-divisors(n)))):
seq(a(n), n = 1..76);

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; f[n_] := f[n] = -DivisorSum[n, V[n, #] * f[#] &, # < n &]; f[1] = 1; a[n_] := Exp[DivisorSum[n, f[#] * Log[n/#] &]]; Array[a, 100] (* Amiram Eldar, Apr 29 2025 *)

def A382941(n: int) -> int: return simplify(exp(sum(A382883(d)*log(n//d) for d in n.divisors())))
print([A382941(n) for n in srange(1, 76)]);

Restricting the sum to prime divisors of n gives A205959(n).
a(k) = 1 <=> k in A000469 <=> k is a nonprime squarefree number.
a(k) != 1 <=> k in A383263 <=> k is prime or divisible by a square greater than 1.

A304404 If n = Product (p_j^k_j) then a(n) = Product (n/p_j^k_j).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 900, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 1764, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 3600, 1, 62, 63, 1, 65, 4356, 1, 68, 69, 4900, 1, 72, 1, 74, 75

Offset: 1

Ilya Gutkovskiy, May 12 2018

nonn

			a(60) = a(2^2*3*5) = (60/2^2) * (60/3) * (60/5) = 15 * 20 * 12 = 3600.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Ilya Gutkovskiy, Logarithmic scatter plot of a(n) up to n=30000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000961 (positions of ones), A001221, A003557, A007774 (fixed points), A028234, A028236, A051119, A062509, A066504, A069359, A072195, A141809, A205959, A284600, A290480.

a[n_] := Times @@ (n/#[[1]]^#[[2]] & /@ FactorInteger[n]); Table[a[n], {n, 75}]
Table[n^(PrimeNu[n] - 1), {n, 75}]

A304404(n) = (n^(omega(n)-1)); \\ Antti Karttunen, Aug 06 2018

from sympy.ntheory.factor_ import primenu
def A304404(n): return int(n**(primenu(n)-1)) # Chai Wah Wu, Jul 12 2023

a(n) = n^(omega(n)-1), where omega() = A001221.

a(n) = A062509(n)/n.

A363918 a(n) = Product_{p in Factors(n)} mult(p)*n^(mult(p) - 1), where Factors(n) is the integer factorization of n and mult(p) the multiplicity of the prime factor p.

Original entry on oeis.org

1, 1, 1, 8, 1, 1, 1, 192, 18, 1, 1, 24, 1, 1, 1, 16384, 1, 36, 1, 40, 1, 1, 1, 1728, 50, 1, 2187, 56, 1, 1, 1, 5242880, 1, 1, 1, 5184, 1, 1, 1, 4800, 1, 1, 1, 88, 90, 1, 1, 442368, 98, 100, 1, 104, 1, 8748, 1, 9408, 1, 1, 1, 120, 1, 1, 126, 6442450944, 1, 1, 1, 136

Offset: 1

Peter Luschny, Jul 19 2023

nonn

Cf. A363919, A363923, A005361, A205959, A303915, A363917.

a := n -> local p: mul(p[2] * n^(p[2] - 1), p in ifactors(n)[2]):
seq(a(n), n = 1..68);

a(n) = my(f=factor(n)[, 2]); vecprod(f)*n^(vecsum(f)-#f); \\ Michel Marcus, Jul 19 2023

a(n) / A363919(n) = A005361(n).

a(n) * A205959(n) = A005361(n) * A363923(n) = A363917(n).

A383438 a(n) = Sum_{k=1..n} Product_{p|k, p prime} k/p.

Original entry on oeis.org

1, 2, 3, 5, 6, 12, 13, 17, 20, 30, 31, 55, 56, 70, 85, 93, 94, 148, 149, 189, 210, 232, 233, 329, 334, 360, 369, 425, 426, 1326, 1327, 1343, 1376, 1410, 1445, 1661, 1662, 1700, 1739, 1899, 1900, 3664, 3665, 3753, 3888, 3934, 3935, 4319, 4326, 4576, 4627, 4731

Offset: 1

Peter Luschny, Apr 27 2025

nonn

Cf. A205959.

a[n_]:=Sum[Product[k/p,{p,Select[Divisors[k],PrimeQ[#] &]}],{k,n}]; Array[a,52] (* Stefano Spezia, Apr 27 2025 *)

a(n) = sum(k=1, n, my(f=factor(k)[, 1]); prod(i=1, #f, k/f[i])); \\ Michel Marcus, Apr 27 2025

from itertools import accumulate
def A383438List(len: int) -> list[int]:
    return list(accumulate([A205959(n) for n in range(1, len + 1)]))
print(A383438List(52))

a(n) = Sum_{k=1..n} A205959(k).

cp's OEIS Frontend

A205957 a(n) = exp(-Sum_{k=1..n} Sum_{d|k, d prime} moebius(d)*log(k/d)).

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A363923 a(n) = n^npf(n) / rad(n), where npf(n) is the number of prime factors with multiplicity of n.

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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).

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A363919 a(n) = n^excess(n), where excess(n) = A046660(n).

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A205958 a(0) = 1 and a(n) = A180000(n)*a(floor(n/2))^2 for n > 0.

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A363917 a(n) = Product_{p in Factors(n)} mult(p) * n^mult(p) / p, where Factors(n) is the integer factorization of n and mult(p) the multiplicity of the prime factor p.

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A382941 a(n) = exp(Sum_{d|n} A382883(d)*log(n/d)).

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A304404 If n = Product (p_j^k_j) then a(n) = Product (n/p_j^k_j).