A063919 Sum of proper unitary divisors (or unitary aliquot parts) of n, including 1.

1, 1, 1, 1, 1, 6, 1, 1, 1, 8, 1, 8, 1, 10, 9, 1, 1, 12, 1, 10, 11, 14, 1, 12, 1, 16, 1, 12, 1, 42, 1, 1, 15, 20, 13, 14, 1, 22, 17, 14, 1, 54, 1, 16, 15, 26, 1, 20, 1, 28, 21, 18, 1, 30, 17, 16, 23, 32, 1, 60, 1, 34, 17, 1, 19, 78, 1, 22, 27, 74, 1, 18, 1, 40, 29, 24, 19, 90, 1, 22, 1, 44

Offset: 1

Author

Felice Russo, Aug 31 2001

Keywords

• easy
• nonn
• nice

Comments

For definition of unitary divisor see A034448.

Examples

			a(10) = 8 because the unitary divisors of 10 are 1, 2, 5 and 10, with sum 18 and 18-10 = 8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Harry J. Smith)

Crossrefs

The values of sequence are A034448(n)-n (for n > 1).

Cf. A001065, A034448, A034460.

Programs

Haskell

a063919 1 = 1
a063919 n = sum $ init $ a077610_row n
-- Reinhard Zumkeller, Mar 12 2012

Maple

A063919 := proc(n)
    if n = 1 then
        1;
    else
        A034448(n)-n ;
    end if;
end proc: # R. J. Mathar, May 14 2013

Mathematica

a[n_] := Total[Select[Divisors[n], GCD[#, n/#] == 1&]]-n; a[1] = 1; Table[a[n], {n, 82}] (* Jean-François Alcover, Aug 31 2011 *)

PARI

usigma(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))
{ for (n=1, 1000, if (n>1, a=usigma(n) - n, a=1); write("b063919.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 02 2009

PARI

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A063919(n) = if(1==n,n,A034460(n)); \\ Antti Karttunen, Jun 12 2018

Formula

a(n) = A034460(n), n>1. - R. J. Mathar, Oct 02 2008

For n > 1: a(n) = sum (A077610(n,k): k = 1 .. A034444(n) - 1). - Reinhard Zumkeller, Mar 12 2012