This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105210 #22 Jul 22 2025 06:00:37
%S A105210 393,528,545,660,682,727,728,751,752,802,1206,1279,1280,1288,1321,
%T A105210 1322,1986,2323,2448,2471,2832,2897,2898,2934,3103,3240,3251,3252,
%U A105210 3529,3530,3891,5192,5265,5287,5616,5635,5671,5832,5838,5990,6597,7334,7549,7550
%N A105210 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 A105210 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 A105210 This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.
%H A105210 T. D. Noe, <a href="/A105210/b105210.txt">Table of n, a(n) for n = 1..2000</a>
%H A105210 Doug Engel, <a href="http://www.jstor.org/stable/2689298">Problem 886</a>, Math. Mag., 48 (1975), 57-58.
%e A105210 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 A105210 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 A105210 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 A105210 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 A105210 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 A105210 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 A105210 nxt[n_]:=n+1+Total[Select[FactorInteger[n][[All,1]],#<n&]]; NestList[ nxt,393,50] (* _Harvey P. Dale_, Mar 02 2019 *)
%o A105210 (Haskell)
%o A105210 a105210 n = a105210_list !! (n-1)
%o A105210 a105210_list = 393 : map
%o A105210 (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a105210_list
%o A105210 -- _Reinhard Zumkeller_, Jan 15 2015
%Y A105210 Cf. A003508, A027748, A105211, A105212, A105213.
%K A105210 nonn,easy
%O A105210 1,1
%A A105210 _R. K. Guy_, Apr 14 2005
%E A105210 More terms from _Robert G. Wilson v_ and _Emeric Deutsch_, Apr 14 2005