A048991 Write down the numbers 1,2,3,... but omit any number (such as 12 or 23 or 31 ...) which appears in the concatenation of all earlier terms.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 32, 33, 35, 36, 37, 38, 39, 40, 43, 44, 46, 47, 48, 49, 50, 54, 55, 57, 58, 59, 60, 65, 66, 68, 69, 70, 76, 77, 79, 80, 87, 88, 90, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112

Offset: 1

Author

Bernardo Recamán

Keywords

• nonn
• nice
• base
• easy

Comments

Similar to the "punctual birds" A131881, numbers which do not occur earlier than their "natural" position in the string 123...9101112..., but more complex to compute due to the fact that here the omitted numbers must be taken into account. Contains powers of ten A011557 as a subsequence, at indices (1, 10, 63, 402, 2954, ...), giving the number of n-digit terms as (9, 53, 339, 2552, ...). - M. F. Hasler, Oct 25 2019

Examples

			12 is omitted since we see "1,2" at the beginning of the sequence; 101 is omitted because we can see "10,1[1]"; etc.
Since 12 is omitted, 21 does not occur "earlier" and it is in this sequence, but not in A131881, since it occurs earlier in "12,13". - _M. F. Hasler_, Oct 25 2019
		

References

• Invented by 10-year-old Hannah Rollman.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Nick Hobson, Python program for this sequence

Crossrefs

See A048992 for the omitted numbers.

Cf. A105390.

Programs

Haskell

import Data.List (isInfixOf)
a048991 n = a048991_list !! (n-1)
a048991_list = f [1..] [] where
   f (x:xs) ys | xs' `isInfixOf` ys = f xs ys
               | otherwise          = x : f xs (xs' ++ ys)
               where xs' = reverse $ show x
-- Reinhard Zumkeller, Dec 05 2011

Mathematica

Clear[a]; a[1] = 1; s = "1"; a[n_] := a[n] = Catch[ For[k = a[n-1] + 1, True, k++, If[ StringFreeQ[s, t = ToString[k]], s = s <> t; Throw[k] ] ] ]; Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Jan 09 2013 *)

PARI

D=[]; for(n=1,199, for(i=0,#D-#d=digits(n), D[i+1..i+#d]==d && next(2)); print1(n","); D=concat(D,d)) \\ M. F. Hasler, Oct 25 2019

Python

# see Hobson link

Extensions

Edited by Patrick De Geest, Jun 02 2003