A034460 a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).

0, 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

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Examples

			Unitary divisors of 12 are 1, 3, 4, 12. a(12) = 1 + 3 + 4 = 8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
• Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Mathematics of Computation 83.288 (2014): 1903-1913; alternative link.

Crossrefs

Cf. A034444, A034448, A077610, A306633.

Cf. A063936 (squares > 1).

Cf. A063919 (essentially the same sequence).

Programs

Haskell

a034460 = sum . init . a077610_row  -- Reinhard Zumkeller, Aug 15 2012

Maple

A034460 := proc(n)
    A034448(n)-n ;
end proc:
seq(A034460(n),n=1..40) ; # R. J. Mathar, Nov 10 2014

Mathematica

usigma[n_] := Sum[ If[GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; a[n_] := usigma[n] - n; Table[ a[n], {n, 1, 82}] (* Jean-François Alcover, May 15 2012 *)
a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 03 2022 *)

PARI

a(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n \\ Charles R Greathouse IV, Aug 01 2016

Formula

a(n) = Sum_{k = 1..A034444(n)-1} A077610(n,k). - Reinhard Zumkeller, Aug 15 2012

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/2 = 0.1842163888... . - Amiram Eldar, Feb 22 2024