OFFSET
1,2
COMMENTS
Also sum of divisors of the squarefree kernel of n: a(n) = A000203(A007947(n)). - Reinhard Zumkeller, Jul 19 2002
The absolute values of the Dirichlet inverse of A001615. - R. J. Mathar, Dec 22 2010
Row sums of the triangle in A206778. - Reinhard Zumkeller, Feb 12 2012
Inverse Möbius transform of n * mu(n)^2 = |A055615(n)|. - Wesley Ivan Hurt, Jun 08 2023
REFERENCES
D. Suryanarayana, On the core of an integer, Indian J. Math. 14 (1972) 65-74.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Steven R. Finch, Unitarism and infinitarism.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
FORMULA
If n = Product p_i^e_i, a(n) = Product (p_i + 1). - Vladeta Jovovic, Apr 19 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(2*s-2). - Michael Somos, Sep 08 2002
a(n) = Sum_{d|n} mu(d)^2*d = Sum_{d|n} |A055615(d)|. - Benoit Cloitre, Dec 09 2002
Pieter Moree (moree(AT)mpim-bonn.mpg.de), Feb 20 2004 can show that Sum_{n <= x} a(n) = x^2/2 + O(x*sqrt{x}) and adds: "As S. R. Finch pointed out to me, in Suryanarayana's paper this is proved under the Riemann hypothesis with error term O(x^{7/5+epsilon})".
G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = 1. - Amiram Eldar, Jun 10 2020
a(n) = Sum_{d divides n} mu(d)^2*core(d), where core(n) = A007913(n). - Peter Bala, Jan 24 2024
EXAMPLE
For n=1000, out of the 16 divisors, four are squarefree: {1,2,5,10}. Their sum is 18. Or, 1000 = 2^3*5^3 hence a(1000) = (2+1)*(5+1) = 18.
MAPLE
A048250 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ] [ 1 ]): od: RETURN(ans) end:
# alternative:
seq(mul(1+p, p = numtheory:-factorset(n)), n=1..1000); # Robert Israel, Mar 18 2015
MATHEMATICA
sumOfSquareFreeDivisors[ n_ ] := Plus @@ Select[ Divisors[ n ], MoebiusMu[ # ] != 0 & ]; Table[ sumOfSquareFreeDivisors[ i ], {i, 85} ]
Table[Total[Select[Divisors[n], SquareFreeQ]], {n, 80}] (* Harvey P. Dale, Jan 25 2013 *)
a[1] = 1; a[n_] := Times@@(1 + FactorInteger[n][[;; , 1]]); Array[a, 100] (* Amiram Eldar, Dec 19 2018 *)
PROG
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)))
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, (1+p*X)/(1-X))[n])
(PARI) a(n)=sumdiv(n, d, moebius(d)^2*d); \\ Joerg Arndt, Jul 06 2011
(PARI) a(n)=my(f=factor(n)); for(i=1, #f~, f[i, 2]=1); sigma(f) \\ Charles R Greathouse IV, Sep 09 2014
(Haskell)
a034448 = sum . a206778_row -- Reinhard Zumkeller, Feb 12 2012
(Sage)
def A048250(n): return mul(map(lambda p: p+1, prime_divisors(n)))
[A048250(n) for n in (1..73)] # Peter Luschny, May 23 2013
(Python)
from math import prod
from sympy import primefactors
def A048250(n): return prod(p+1 for p in primefactors(n)) # Chai Wah Wu, Apr 20 2023
CROSSREFS
KEYWORD
nonn,easy,nice,mult
AUTHOR
STATUS
approved