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a(n) = sigma(n) + usigma(n) = A000203(n) + A034448(n).
2

%I #18 Sep 03 2024 14:54:11

%S 2,6,8,12,12,24,16,24,23,36,24,48,28,48,48,48,36,69,40,72,64,72,48,96,

%T 57,84,68,96,60,144,64,96,96,108,96,141,76,120,112,144,84,192,88,144,

%U 138,144,96,192,107,171,144,168,108,204,144,192,160

%N a(n) = sigma(n) + usigma(n) = A000203(n) + A034448(n).

%H Harry J. Smith, <a href="/A064212/b064212.txt">Table of n, a(n) for n = 1..1000</a>

%F Sum_{k=1..n} a(k) ~ (Pi^2/12)*(1 + 1/zeta(3)) * n^2. - _Amiram Eldar_, Apr 01 2024

%t f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) + Times @@ f2 @@@ fct; a[1] = 2; Array[a, 100] (* _Amiram Eldar_, Apr 01 2024 *)

%o (PARI) usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }

%o { for (n=1, 1000, write("b064212.txt", n, " ", sigma(n) + usigma(n)) ) } \\ _Harry J. Smith_, Sep 10 2009

%Y Cf. A000203, A034448, A048146.

%Y Cf. A002117, A072691.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Oct 31 2001