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%I #20 Mar 02 2019 11:08:30
%S 393,528,545,660,682,727,728,751,752,802,1206,1279,1280,1288,1321,
%T 1322,1986,2323,2448,2471,2832,2897,2898,2934,3103,3240,3251,3252,
%U 3529,3530,3891,5192,5265,5287,5616,5635,5671,5832,5838,5990,6597,7334,7549,7550
%N a(1) = 393; for n > 1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1). edit
%C 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.
%C This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.
%H T. D. Noe, <a href="/A105210/b105210.txt">Table of n, a(n) for n = 1..2000</a>
%H Doug Engel, <a href="http://www.jstor.org/stable/2689298">Problem 886</a>, Math. Mag., 48 (1975), 57-58.
%e a(2)=528 because a(1)=393, the distinct prime factors of a(1) are 3 and 131; finally, 1 + 393 + 3 + 131 = 528.
%p 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]:=393: 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 a[1] = 393; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[a[n - 1]]], # < a[n - 1] &]; Table[a[n], {n, 44}] (* _Robert G. Wilson v_, Apr 14 2005 *)
%t a[1] = 412; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[a[n - 1]]], # < a[n - 1] &]; Table[a[n], {n, 43}] (* _Robert G. Wilson v_, Apr 14 2005 *)
%t a[1] = 668; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[a[n - 1]]], # < a[n - 1] &]; Table[a[n], {n, 40}] (* _Robert G. Wilson v_, Apr 14 2005 *)
%t a[1] = 932; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[a[n - 1]]], # < a[n - 1] &]; Table[a[n], {n, 40}] (* _Robert G. Wilson v_, Apr 14 2005 *)
%t nxt[n_]:=n+1+Total[Select[FactorInteger[n][[All,1]],#<n&]]; NestList[ nxt,393,50] (* _Harvey P. Dale_, Mar 02 2019 *)
%o (Haskell)
%o a105210 n = a105210_list !! (n-1)
%o a105210_list = 393 : map
%o (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a105210_list
%o -- _Reinhard Zumkeller_, Jan 15 2015
%Y Cf. A003508, A027748, A105211, A105212, A105213.
%K nonn,easy
%O 1,1
%A _R. K. Guy_, Apr 14 2005
%E More terms from _Robert G. Wilson v_ and _Emeric Deutsch_, Apr 14 2005
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