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A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1). 115
1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68, 90, 96, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums of the triangle in A077610. [Reinhard Zumkeller, Feb 12 02]

REFERENCES

Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., to appear (2014); http://www.math.dartmouth.edu/~carlp/uupaper7.pdf

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

S. R. Finch, Unitarism and infinitarism.

Eric Weisstein's World of Mathematics, Unitary Divisor Function

Wikipedia, Unitary_divisor

FORMULA

If n = Product p_i^e_i, usigma(n) = Product (p_i^e_i + 1) - Vladeta Jovovic, Apr 19 2001

Dirichlet generating function: zeta(s)*zeta(s-1)/zeta(2s-1). - Franklin T. Adams-Watters, Sep 11 2005.

Multiplicative with a(p^e) = p^e+1 for e>0. - Franklin T. Adams-Watters, Sep 11 2005.

EXAMPLE

Unitary divisors of 12 are 1, 3, 4, 12. Or, 12=3*2^2 hence usigma(12)=(3+1)*(2^2+1)=20.

MAPLE

A034448 := 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 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: RETURN(ans) end:

a := proc(n) local i; numtheory[divisors](n); select(d -> igcd(d, n/d)=1, %); add(i, i=%) end; [From Peter Luschny, May 03 2009]

MATHEMATICA

usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; Table[ usigma[n], {n, 71}] (from Robert G. Wilson v Aug 28 2004)

f[n_] := Block[{d = Divisors@ n}, Plus @@ EulerPhi@ Select[d, GCD[ #, n/# ] == 1 &]]; Array[f, 73] [From Robert G. Wilson v, Sep 09 2008]

PROG

(PARI) A034448(n)=sumdiv(n, d, if(gcd(d, n/d)==1, d)) \\ Rick L. Shepherd

(Haskell)

a034448 = sum . a077610_row  -- Reinhard Zumkeller, Feb 12 2012

CROSSREFS

Cf. A000203, A034444, A034460, A047994, A048250, A064000.

Cf. A063937 (squares > 1).

Cf. A188999.

Sequence in context: A103402 A154664 A191750 * A069184 A181549 A049417

Adjacent sequences:  A034445 A034446 A034447 * A034449 A034450 A034451

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Erich Friedman.

STATUS

approved

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Last modified April 16 18:58 EDT 2014. Contains 240627 sequences.