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a(1) = a(2) = 1; for n >= 2, a(n+1) = a(n) + a(j(n)) where j(m) = largest divisor of m that is < m.
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%I #29 Jun 10 2018 10:31:58

%S 1,1,2,3,4,5,7,8,11,13,17,18,23,24,31,35,43,44,55,56,69,76,93,94,112,

%T 116,139,150,174,175,206,207,242,259,302,309,353,354,409,432,488,489,

%U 558,559,635,666,759,760,854,861,973,1016,1132,1133,1272,1289,1439,1494

%N a(1) = a(2) = 1; for n >= 2, a(n+1) = a(n) + a(j(n)) where j(m) = largest divisor of m that is < m.

%H T. D. Noe, <a href="/A080655/b080655.txt">Table of n, a(n) for n = 1..1000</a>

%t a[1]=a[2]=1; a[n_] := a[n] = a[n-1]+a[Divisors[n-1][[-2]]]; Table[a[n],{n,1,58}] (* _Jean-François Alcover_, Mar 20 2011 *)

%o (PARI) a(n) = if(n==1,1,if(n==2,1,a(n-1)+a((n-1)/divisors(n-1)[2]))) /* j(m) = (largest divisor) = m / (smallest divisor) */ \\ _Michael B. Porter_, Oct 06 2009

%o (Haskell)

%o a080655 n = a080655_list !! (n-1)

%o a080655_list = 1 : 1 : f 3 where

%o f n = (a080655 (n-1) + a080655 (a032742 (n-1))) : f (n+1)

%o -- _Reinhard Zumkeller_, Aug 21 2011

%K nonn,easy,nice

%O 1,3

%A _Leroy Quet_, Feb 28 2003

%E More terms from _Naohiro Nomoto_, Mar 07 2003