A003619 Not of form [ e^m ], m >= 1. (Formerly M2300)

1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53

Offset

1,2

Comments

If 1 is excluded (of form [e^0]) then complement of A000149. - Michel Marcus, Jun 16 2013

References

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 11.

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

J. Lambek and L. Moser, Inverse and complementary sequences of natural numbers, Amer. Math. Monthly, 61 (1954), 454-458.

Formula

a(n) = n + [ log (n + 1 + [ log (n + 1) ]) ].

a(n) = n+m if n+m>=exp(m) and a(n) = n+m-1 otherwise, where m = 1+floor(log(n)). - Chai Wah Wu, Nov 09 2025

Mathematica

Table[n + Floor@ Log[n + 1 + Floor@ Log[n + 1]], {n, 50}] (* Michael De Vlieger, Oct 06 2017 *)

Prog

(PARI) a(n) = n + floor( log (n + 1 + floor( log (n + 1) )) ) \\ Michel Marcus, Jun 16 2013
(Haskell)
a003619 n = n + floor (log (x + fromIntegral (floor $ log x)))
            where x = fromIntegral n + 1
-- Reinhard Zumkeller, Mar 17 2015
(Python)
from sympy import exp, log, floor
def A003619(n): return n+(m:=int(1+floor(log(n))))-bool(n+m<exp(m)) # Chai Wah Wu, Nov 09 2025

Crossrefs

Cf. A000195.

Sequence in context: A298420 A197354 A089399 * A377721 A216846 A288674

Adjacent sequences: A003616 A003617 A003618 * A003620 A003621 A003622

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein

Status

approved