A323669 -id:A323669 - cp's OEIS frontend

A073482 Largest prime factor of the n-th squarefree number.

1, 2, 3, 5, 3, 7, 5, 11, 13, 7, 5, 17, 19, 7, 11, 23, 13, 29, 5, 31, 11, 17, 7, 37, 19, 13, 41, 7, 43, 23, 47, 17, 53, 11, 19, 29, 59, 61, 31, 13, 11, 67, 23, 7, 71, 73, 37, 11, 13, 79, 41, 83, 17, 43, 29, 89, 13, 31, 47, 19, 97, 101, 17, 103, 7, 53, 107, 109, 11, 37, 113, 19, 23, 59, 17, 61, 41

Offset: 1

Reinhard Zumkeller, Aug 03 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Rafael Jakimczuk, Summing the largest prime factor over integer sequences, Revista de la Unión Matemática Argentina, Vol. 67, No. 1 (2024), pp. 27-35.

Cf. A073481, A001221, A005117, A006530, A265668, A323669.

a073482 = a006530 . a005117  -- Reinhard Zumkeller, Feb 04 2012

issquarefree := proc(n::integer) local nf, ifa, lar; nf := op(2,ifactors(n)); for ifa from 1 to nops(nf) do lar := op(1,op(ifa,nf)); if op(2,op(ifa,nf)) >= 2 then RETURN(0); fi; od : RETURN(lar); end: printf("1,"); for n from 2 to 100 do lfa := issquarefree(n); if lfa > 0 then printf("%a,",lfa); fi; od : # R. J. Mathar, Apr 02 2006

FactorInteger[#][[-1, 1]]& /@ Select[Range[100], SquareFreeQ] (* Jean-François Alcover, Feb 01 2018 *)
s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 2 &], f[[-1, 1]], Nothing]]; Array[s, 200] (* Amiram Eldar, Mar 03 2024 *)

do(x)=my(v=List([1])); forfactored(n=2,x\1, if(vecmax(n[2][,2])==1, listput(v, vecmax(n[2][,1])))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

from math import isqrt
from sympy import mobius, primefactors
def A073482(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return max(primefactors(kmax),default=1) # Chai Wah Wu, Aug 28 2024

a(n) = A006530(A005117(n)).
a(n) = A265668(n, A001221(n)). - Reinhard Zumkeller, Dec 13 2015
Sum_{A005117(n) <= x} a(n) = Sum_{i=1..k} d_i * x^2/log(x)^i + O(x^2/log(x)^(k+1)), for any given positive integer k, where d_i are constants, d_1 = 15/(2*Pi^2) = 0.759908... (A323669) (De Koninck and Jakimczuk, 2024). - Amiram Eldar, Mar 03 2024

More terms from Jason Earls, Aug 06 2002

A121176 Union of {8, 9, 18}, S, 2S and 4S, where S = squarefree numbers (A005117).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92

Offset: 1

N. J. A. Sloane, Sep 13 2006

nonn

The asymptotic density of this sequence is 15/(2*Pi^2) (A323669). - Amiram Eldar, May 10 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A121684, A323669.

Union[Join[{9, 18}, Select[Range[100], (e = IntegerExponent[#,2]) <= 3 && SquareFreeQ[#/2^e] &]]] (* Amiram Eldar, May 10 2022 *)

A327340 Numerator of the rationals r(n) = (1/n^2)*Phi_1(n), with Phi_1(n) = Sum{k=1..n} psi(k), with Dedekind's psi function.

Original entry on oeis.org

1, 1, 8, 7, 4, 8, 40, 13, 64, 41, 94, 59, 132, 39, 4, 51, 222, 43, 278, 157, 346, 191, 406, 227, 484, 263, 562, 305, 640, 178, 24, 99, 280, 447, 942, 169, 1052, 278, 1168, 31, 1282, 689, 1422, 747, 58, 819, 1686, 99, 1838, 482

Offset: 1

Wolfdieter Lang, Sep 03 2019

The corresponding denominators are given in A327341.
Dedekind's psi(k) = k*Product_{p|k}(1 + 1/p), with primes p, and the empty product is set to 1. See psi(k) = A001615(k), k >= 1. In the Walfisz reference psi(k) = phi_1(k).
In the  Walfisz reference, Satz 2., p. 100, the approximation for Phi_1(x) = (15/(2*Pi^2))*x^2 + O(x*(log(x))^{2/3}) is given (with B instead of the O() notation). For the constant 15/(2*Pi^2) see A323669 .

			The rationals (in lowest terms) begin: 1/1, 1/1, 8/9, 7/8, 4/5, 8/9, 40/49, 13/16, 64/81, 41/50, 94/121, 59/72, 132/169, 39/49, 4/5, 51/64, 222/289, 43/54, 278/361, 157/200, 346/441, 191/242, 406/529, 227/288, 484/625, 263/338, 562/729, 305/392, 640/841, 178/225, 24/31, ...
The limit of r(n) for n-> infinity is A323669 = 0.759908877317533285829...
r(10^5) is approximatly 0.7599142240 (10 digits).

Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 100, Satz 2.

Eric Weisstein's World of Mathematics, Dedekind Function.
Wikipedia, Dedekind psi function.

Cf. A001615, A173290, A323669, A327341 (denominators).

psi[0] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); a[n_] := Numerator[Sum[psi[k], {k, 1, n}]/n^2]; Array[a, 50] (* Amiram Eldar, Sep 03 2019 *)

dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = numerator(sum(k=1, n, dpsi(k))/n^2); \\ Michel Marcus, Sep 18 2023

a(n) = numerator(r(n)), with the rationals r(n) = (1/n^2)*Sum{k=1..n}(k*Product_{p|k}(1 + 1/p)), with distinct prime p divisors of k (with empty product set to 1 for k = 1), for n >= 1.

a(n) = numerator(A173290(n)/n^2). - Amiram Eldar, Nov 24 2022

A143268 a(n) = phi(n)T(n), where phi(n) is Euler's totient function (A000010) and T(n) = n(n+1)/2 is the n-th triangular number (A000217).

Original entry on oeis.org

1, 3, 12, 20, 60, 42, 168, 144, 270, 220, 660, 312, 1092, 630, 960, 1088, 2448, 1026, 3420, 1680, 2772, 2530, 6072, 2400, 6500, 4212, 6804, 4872, 12180, 3720, 14880, 8448, 11220, 9520, 15120, 7992, 25308, 13338, 18720, 13120, 34440, 10836, 39732

Offset: 1

Gary W. Adamson, Aug 03 2008

nonn

			a(4) = 20 = phi(4) * T(4) = 2 * 10.
a(4) = 20 = sum of row 4 terms of triangle A143267: (2 + 4 + 6 + 8).

Cf. A000010, A000217, A143267, A323669.

with(numtheory): seq((1/2)*n*(n+1)*phi(n),n=1..45); # Emeric Deutsch, Aug 23 2008

a(n)=eulerphi(n)*n*(n+1)/2 \\ Charles R Greathouse IV, Mar 05 2013

a(n) = sum of n-th row of triangle A143267.
a(n) = n*(n+1)*phi(n)/2. - Emeric Deutsch, Aug 23 2008
Sum_{k=1..n} a(k) ~ c * n^4, where c = 3/(4*Pi^2) = A323669 / 10 = 0.0759908... . - Amiram Eldar, Nov 27 2024

Extended by Emeric Deutsch, Aug 16 2008

cp's OEIS Frontend

A073482 Largest prime factor of the n-th squarefree number.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A121176 Union of {8, 9, 18}, S, 2S and 4S, where S = squarefree numbers (A005117).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A327340 Numerator of the rationals r(n) = (1/n^2)*Phi_1(n), with Phi_1(n) = Sum{k=1..n} psi(k), with Dedekind's psi function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A143268 a(n) = phi(n)*T(n), where phi(n) is Euler's totient function (A000010) and T(n) = n*(n+1)/2 is the n-th triangular number (A000217).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A143268 a(n) = phi(n)T(n), where phi(n) is Euler's totient function (A000010) and T(n) = n(n+1)/2 is the n-th triangular number (A000217).