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A105211 a(1) = 412; for n > 1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

%I A105211 #18 Aug 18 2017 03:17:19
%S A105211 412,518,565,684,709,710,789,1056,1073,1140,1170,1194,1399,1400,1415,
%T A105211 1704,1781,1932,1968,2015,2065,2137,2138,3210,3328,3344,3377,3696,
%U A105211 3720,3762,3798,4015,4105,4932,5075,5117,5185,5269,5760,5771,6000,6011,6012
%N A105211 a(1) = 412; for n > 1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).
%C A105211 In Math. Mag. 48 (1975) 301 one finds "C. W. Trigg, C. C. Oursler and R. Cormier and J. L. Selfridge have sent calculations on Problem 886 [Nov 1973] for which we had received only partial results [Jan 1975]. Cormier and Selfridge sent the following results: There appear to be five sequences beginning with integers less than 1000 which do not merge. These sequences were carried out to 10^8 or more." The five sequences are A003508, A105210-A105213.
%H A105211 T. D. Noe, <a href="/A105211/b105211.txt">Table of n, a(n) for n = 1..2000</a>
%H A105211 Doug Engel, <a href="http://www.jstor.org/stable/2689298">Problem 886</a>, Math. Mag., 48 (1975), 57-58.
%e A105211 a(2)=518 because a(1)=412, the distinct prime factors of a(1) are 2 and 103; finally, 1 + 412 + 2 + 103 = 518.
%p A105211 with(numtheory): p:=proc(n) local nn,ct,s: if isprime(n)=true then s:=0 else nn:=convert(factorset(n),list): ct:=nops(nn): s:=sum(nn[j],j=1..ct):fi: end: a[1]:=412: for n from 2 to 50 do a[n]:=1+a[n-1]+p(a[n-1]) od:seq(a[n],n=1..50); # _Emeric Deutsch_, Apr 14 2005
%t A105211 nxt[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#<n&]]; NestList[ nxt,412,50] (* _Harvey P. Dale_, Sep 13 2013 *)
%o A105211 (Haskell)
%o A105211 a105211 n = a105211_list !! (n-1)
%o A105211 a105211_list = 412 : map
%o A105211       (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a105211_list
%o A105211 -- _Reinhard Zumkeller_, Jan 15 2015
%Y A105211 Cf. A003508, A027748, A105210, A105212, A105213.
%K A105211 nonn,easy
%O A105211 1,1
%A A105211 _R. K. Guy_, Apr 14 2005
%E A105211 More terms from _Robert G. Wilson v_ and _Emeric Deutsch_, Apr 14 2005