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.

393, 528, 545, 660, 682, 727, 728, 751, 752, 802, 1206, 1279, 1280, 1288, 1321, 1322, 1986, 2323, 2448, 2471, 2832, 2897, 2898, 2934, 3103, 3240, 3251, 3252, 3529, 3530, 3891, 5192, 5265, 5287, 5616, 5635, 5671, 5832, 5838, 5990, 6597, 7334, 7549, 7550

Offset: 1

Author

R. K. Guy, Apr 14 2005

Keywords

• nonn
• easy

Comments

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.

This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.

Examples

			a(2)=528 because a(1)=393, the distinct prime factors of a(1) are 3 and 131; finally, 1 + 393 + 3 + 131 = 528.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..2000
• Doug Engel, Problem 886, Math. Mag., 48 (1975), 57-58.

Crossrefs

Cf. A003508, A027748, A105211, A105212, A105213.

Programs

Haskell

a105210 n = a105210_list !! (n-1)
a105210_list = 393 : map
      (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a105210_list
-- Reinhard Zumkeller, Jan 15 2015

Maple

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

Mathematica

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 *)
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 *)
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 *)
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 *)
nxt[n_]:=n+1+Total[Select[FactorInteger[n][[All,1]],#Harvey P. Dale, Mar 02 2019 *)

Extensions

More terms from Robert G. Wilson v and Emeric Deutsch, Apr 14 2005