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%I #41 Feb 22 2024 02:15:30
%S 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,
%T 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,
%U 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
%N a(n) = usigma(n) - n, where usigma(n) = sum of unitary divisors of n (A034448).
%H Antti Karttunen, <a href="/A034460/b034460.txt">Table of n, a(n) for n = 1..65537</a> (first 1000 terms from T. D. Noe)
%H Carl Pomerance and Hee-Sung Yang, <a href="https://doi.org/10.1090/S0025-5718-2013-02775-5">Variant of a theorem of Erdős on the sum-of-proper-divisors function</a>, Mathematics of Computation 83.288 (2014): 1903-1913; <a href="http://www.math.dartmouth.edu/~carlp/uupaper7.pdf">alternative link</a>.
%F a(n) = Sum_{k = 1..A034444(n)-1} A077610(n,k). - _Reinhard Zumkeller_, Aug 15 2012
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/2 = 0.1842163888... . - _Amiram Eldar_, Feb 22 2024
%e Unitary divisors of 12 are 1, 3, 4, 12. a(12) = 1 + 3 + 4 = 8.
%p A034460 := proc(n)
%p A034448(n)-n ;
%p end proc:
%p seq(A034460(n),n=1..40) ; # _R. J. Mathar_, Nov 10 2014
%t 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 *)
%t a[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - n; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Oct 03 2022 *)
%o (Haskell)
%o a034460 = sum . init . a077610_row -- _Reinhard Zumkeller_, Aug 15 2012
%o (PARI) a(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n \\ _Charles R Greathouse IV_, Aug 01 2016
%Y Cf. A034444, A034448, A077610, A306633.
%Y Cf. A063936 (squares > 1).
%Y Cf. A063919 (essentially the same sequence).
%K nonn,easy,nice
%O 1,6
%A _N. J. A. Sloane_
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