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A048250 Sum of squarefree divisors of n. 29
1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 12, 14, 24, 24, 3, 18, 12, 20, 18, 32, 36, 24, 12, 6, 42, 4, 24, 30, 72, 32, 3, 48, 54, 48, 12, 38, 60, 56, 18, 42, 96, 44, 36, 24, 72, 48, 12, 8, 18, 72, 42, 54, 12, 72, 24, 80, 90, 60, 72, 62, 96, 32, 3, 84, 144, 68, 54, 96, 144, 72, 12, 74 (list; graph; refs; listen; history; text; internal format)
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

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

S. R. Finch, Unitarism and infinitarism.

Index entries for sequences related to sums of squares

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. - 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})".

a(n) = psi(rad(n)) = A001615(A007947(n)). - Enrique Pérez Herrero, Aug 24 2010

a(n) = rad(n)*psi(n)/n = A001615(n)*A007947(n)/n. - Enrique Pérez Herrero, Aug 31 2010

G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017

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 *)

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

CROSSREFS

Cf. A003557, A007947, A023900, A034444, A034448, A206787.

Sequence in context: A061800 A218789 A238162 * A073181 A183100 A046897

Adjacent sequences:  A048247 A048248 A048249 * A048251 A048252 A048253

KEYWORD

nonn,easy,nice,mult

AUTHOR

Labos Elemer

STATUS

approved

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Last modified May 23 08:55 EDT 2017. Contains 286909 sequences.