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A371061 a(1)=1, a(2)=2; for n > 2, a(n) is the sum of the largest proper divisor of each of the previous two terms, except that the term itself is used if it has no proper divisors > 1.

%I A371061 #37 May 25 2024 15:31:52
%S A371061 1,2,3,5,8,9,7,10,12,11,17,28,31,45,46,38,42,40,41,61,102,112,107,163,
%T A371061 270,298,284,291,239,336,407,205,78,80,79,119,96,65,61,74,98,86,92,89,
%U A371061 135,134,112,123,97,138,166,152,159,129,96,91,61,74,98
%N A371061 a(1)=1, a(2)=2; for n > 2, a(n) is the sum of the largest proper divisor of each of the previous two terms, except that the term itself is used if it has no proper divisors > 1.
%H A371061 Paolo Xausa, <a href="/A371061/b371061.txt">Table of n, a(n) for n = 1..10000</a>
%H A371061 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1).
%F A371061 a(n) = A117818(a(n-1)) + A117818(a(n-2)) for n > 2.
%F A371061 a(n+18) = a(n) for n >= 39. - _R. J. Mathar_, May 24 2024
%e A371061 Each term is the sum of the largest proper divisors of the previous two terms. If a term has no proper divisors > 1 then take the number itself, e.g.:
%e A371061 a(2) = 2 is prime, a(3) = 3 is prime, so a(4) = 2+3 = 5;
%e A371061 a(3) = 3 is prime, a(4) = 5 is prime, so a(5) = 3+5 = 8;
%e A371061 a(4) = 5 is prime, a(5) = 8, whose largest proper divisor is 4, so a(6) = 5+4 = 9;
%e A371061 the largest proper divisors of 8 and 9 are 4 and 3, respectively, so a(7) = 4+3 = 7; etc.
%p A371061 A371061 := proc(n)
%p A371061     option remember ;
%p A371061     if n <= 2 then
%p A371061         n;
%p A371061     else
%p A371061         A117818(procname(n-1))+A117818(procname(n-2)) ;
%p A371061     end if;
%p A371061 end proc:
%p A371061 seq(A371061(n),n=1..100) ; # _R. J. Mathar_, Apr 30 2024
%t A371061 A117818[n_] := If[n == 1 || PrimeQ[n], n, Divisors[n][[-2]]];
%t A371061 A371061[n_] := A371061[n] = If[n < 3, n, A117818[A371061[n-1]] + A117818[A371061[n-2]]];
%t A371061 Array[A371061, 100] (* _Paolo Xausa_, May 24 2024 *)
%o A371061 (Python)
%o A371061 import math
%o A371061 def primeTest(num):
%o A371061     ans = 0
%o A371061     if num == 1 or num == 2:
%o A371061         return num
%o A371061     for i in range(0,int(num**0.5)):
%o A371061         if (num/(i+2)).is_integer() == True:
%o A371061             ans = num/(i+2)
%o A371061             break
%o A371061     if ans == 0:
%o A371061         ans = num
%o A371061     return ans
%o A371061 def seqgen(start):
%o A371061     seq = start
%o A371061     x = [0,1]
%o A371061     for i in range(0,1000):
%o A371061         list = ''.join(str(x) for x in seq)
%o A371061         a = primeTest(seq[i])
%o A371061         b = primeTest(seq[i+1])
%o A371061         seq.append(int(a+b))
%o A371061         x.append(i+2)
%o A371061         sublist = ''.join(str(x) for x in [seq[i],seq[i+1],seq[i+2]])
%o A371061         if sublist in list:
%o A371061             break
%o A371061     return i,x,seq
%o A371061 start = [1,2]
%o A371061 i,x,seq = seqgen(start)
%o A371061 print(seq)
%o A371061 (PARI) \\ b(n) is A117818(n).
%o A371061 b(n)=if(n==1 || isprime(n), n, n/factor(n)[1,1])
%o A371061 seq(n) = {my(a=vector(n)); a[1]=1; a[2]=2; for(n=3, n, a[n] = b(a[n-1]) + b(a[n-2])); a} \\ _Andrew Howroyd_, Mar 09 2024
%Y A371061 Cf. A117818.
%K A371061 nonn,easy
%O A371061 1,2
%A A371061 _Thomas W. Young_, Mar 09 2024