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A249357 Fibonacci-Zumkeller numbers: a(n)=n if n<=3, otherwise the smallest number >= a(n-2) + a(n-1) having at least one common factor with a(n-2), but none with a(n-1).

%I A249357 #49 Jan 26 2025 09:12:09
%S A249357 1,2,3,8,15,26,45,74,123,200,327,530,861,1396,2259,3656,5919,9578,
%T A249357 15501,25082,40587,65672,106263,171938,278211,450151,728367,1178527,
%U A249357 1906896,3085439,4992336,8077783,13070121,21147910,34218033,55365944,89583981,144949928,234533913,379483844,614017761,993501608
%N A249357 Fibonacci-Zumkeller numbers: a(n)=n if n<=3, otherwise the smallest number >= a(n-2) + a(n-1) having at least one common factor with a(n-2), but none with a(n-1).
%C A249357 To construct Fibonacci-like sequence, we use a rule from the definition of A098550.
%H A249357 Chai Wah Wu, <a href="/A249357/b249357.txt">Table of n, a(n) for n = 1..500</a>
%e A249357 a(3)+a(4)=3+8=11. However, gcd(11,3)=1, further, gcd(12,8)>1, gcd(13,3)=1, gcd(14,8)>1, finally, gcd(15,3)>1 and gcd(15,8)=1. Thus 15 is the smallest number >11 which satisfies the definition. So a(5)=15.
%p A249357 for n from 1 to 3 do a[n]:= n od:
%p A249357 for n from 4 to 100 do
%p A249357   for k from a[n-1]+a[n-2] do
%p A249357     if igcd(k,a[n-2]) > 1 and igcd(k,a[n-1]) = 1 then
%p A249357        a[n]:= k;
%p A249357        break
%p A249357     fi
%p A249357   od
%p A249357 od:
%p A249357 seq(a[n],n=1..100); # _Robert Israel_, Dec 03 2014
%t A249357 A249357={1,2,3};Do[AppendTo[A249357,NestWhile[#+1&,A249357[[-1]]+A249357[[-2]],!(GCD[#,A249357[[-1]]]==1&&GCD[#,A249357[[-2]]]>1)&]],{50}];A249357 (* _Peter J. C. Moses_, Dec 03 2014 *)
%o A249357 (PARI) a(n, show=1, a=3, o=2)={n<3&&return(n); show&&print1("1,2"); for(i=4,n, show&&print1(","a); k=a+o; until(gcd(k,o)>1 && gcd(k,a)==1,k++); o=a; a=k); a} \\ _M. F. Hasler_, Dec 03 2014
%o A249357 (Python)
%o A249357 from math import gcd
%o A249357 A249357_list, l1, l2 = [1,2,3], 3, 2
%o A249357 for _ in range(100):
%o A249357     i = l1+l2
%o A249357     while True:
%o A249357         if gcd(i,l1) == 1 and gcd(i,l2) > 1:
%o A249357             A249357_list.append(i)
%o A249357             l2, l1 = l1, i
%o A249357             break
%o A249357         i += 1 # _Chai Wah Wu_, Dec 04 2014
%o A249357 (Haskell)
%o A249357 a249357 n = a249357_list
%o A249357 a249357_list = 1 : 2 : 3 : f 2 3 where
%o A249357    f u v = y : f v y where
%o A249357      y = head [x | x <- [u + v ..], gcd x u > 1, gcd x v == 1]
%o A249357 -- _Reinhard Zumkeller_, Dec 04 2014
%Y A249357 Cf. A000045, A098550, A251608.
%K A249357 nonn
%O A249357 1,2
%A A249357 _Vladimir Shevelev_, Dec 03 2014
%E A249357 More terms from _M. F. Hasler_, Dec 03 2014