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a(1) = 1; for n>1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).
(Formerly M0580)
10

%I M0580 #28 Aug 18 2017 03:21:00

%S 1,2,3,4,7,8,11,12,18,24,30,41,42,55,72,78,97,98,108,114,139,140,155,

%T 192,198,215,264,281,282,335,408,431,432,438,517,576,582,685,828,857,

%U 858,888,931,958,1440,1451,1452,1469,1596,1628,1679,1776,1819,1944

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

%C R. K. Guy reports, Apr 14 2005: In Math. Mag. 48 (1975) 301 one finds "C. W. Trigg, C. C. Oursler and R. Cormier & 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.

%C All terms of these five sequences are distinct up to least 10^30. - _T. D. Noe_, Oct 19 2007

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A003508/b003508.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(6)=8, so a(7) = 8 + 1 + 2 = 11.

%t a[1] = 1; 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, 54}] (* _Robert G. Wilson v_, Apr 13 2005 *)

%t nxt[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#<n&]]; NestList[ nxt,1,60] (* _Harvey P. Dale_, Jul 19 2015 *)

%o (Haskell)

%o a003508 n = a003508_list !! (n-1)

%o a003508_list = 1 : map

%o (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a003508_list

%o -- _Reinhard Zumkeller_, Jan 15 2015

%Y Cf. A096460, A105221, A105233.

%Y Cf. A027748, A105210, A105211, A105212, A105213.

%K nonn,nice,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Henry Bottomley_, May 09 2000