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A034448 usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1). 120
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 2002

LINKS

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

S. R. Finch, Unitarism and infinitarism.

Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., to appear (2014);

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; # Peter Luschny, May 03 2009

MATHEMATICA

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

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

PROG

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

(PARI) A034448(n) = {my(f=factorint(n)); prod(k=1, #f[, 2], f[k, 1]^f[k, 2]+1)} \\ Andrew Lelechenko, Apr 22 2014

(PARI) a(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d)) \\ Charles R Greathouse IV, Sep 09 2014

(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 A241405

Adjacent sequences:  A034445 A034446 A034447 * A034449 A034450 A034451

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane, Dec 11 1999

EXTENSIONS

More terms from Erich Friedman

STATUS

approved

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Last modified December 22 15:37 EST 2014. Contains 252364 sequences.