A003508 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).

1, 2, 3, 4, 7, 8, 11, 12, 18, 24, 30, 41, 42, 55, 72, 78, 97, 98, 108, 114, 139, 140, 155, 192, 198, 215, 264, 281, 282, 335, 408, 431, 432, 438, 517, 576, 582, 685, 828, 857, 858, 888, 931, 958, 1440, 1451, 1452, 1469, 1596, 1628, 1679, 1776, 1819, 1944

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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.

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

All terms of these five sequences are distinct up to least 10^30. - T. D. Noe, Oct 19 2007

Examples

			a(6)=8, so a(7) = 8 + 1 + 2 = 11.
		

References

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

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. A096460, A105221, A105233.

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

Programs

Haskell

a003508 n = a003508_list !! (n-1)
a003508_list = 1 : map
      (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a003508_list
-- Reinhard Zumkeller, Jan 15 2015

Mathematica

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 *)
nxt[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#Harvey P. Dale, Jul 19 2015 *)

Extensions

More terms from Henry Bottomley, May 09 2000