A105390 -id:A105390 - cp's OEIS frontend

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

Bernardo Recamán

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

			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

Invented by 10-year-old Hannah Rollman.

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

See A048992 for the omitted numbers.

Cf. A105390.

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

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 *)

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
```

Edited by Patrick De Geest, Jun 02 2003

A048992 Hannah Rollman's numbers: the numbers excluded from A048991.

Original entry on oeis.org

12, 23, 31, 34, 41, 42, 45, 51, 52, 53, 56, 61, 62, 63, 64, 67, 71, 72, 73, 74, 75, 78, 81, 82, 83, 84, 85, 86, 89, 91, 92, 93, 94, 95, 96, 97, 98, 101, 111, 113, 121, 122, 123, 131, 141, 151, 161, 171, 181, 191, 192, 201, 202, 210, 211, 212, 213, 214, 215, 216, 217

Offset: 1

Bernardo Recamán

A105390(n) = number of terms <= n; for n < 740: A105390(n) < n/2. - Reinhard Zumkeller, Apr 04 2005
A116700 is a similar sequence. Note that 21 is missing from the current sequence, because we deleted 12 in computing A048991 and now 21 is no longer "earlier in the sequence". On the other hand 21 is present in A116700. - N. J. A. Sloane, Aug 05 2007
Otherwise said: Numbers which occur in the concatenation of all smaller numbers not listed in this sequence. - M. F. Hasler, Dec 29 2012
Number of terms < 10^n, n = 1, 2, ...: (0, 37, 589, 7046, ...), gives number of n-digit terms as first differences: (37, 552, 6457, ...). - M. F. Hasler, Oct 25 2019

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

Complement of A048991.

Similar to A116700: "early birds" in the Barbier word A007376 or Champernowne sequence A033307.

import Data.List (isInfixOf)
a048992 n = a048992_list !! (n-1)
a048992_list = g [1..] [] where
   g (x:xs) ys | xs' `isInfixOf` ys = x : g xs ys
               | otherwise          = g xs (xs' ++ ys)
               where xs' = reverse $ show x
-- Reinhard Zumkeller, Dec 05 2011

a[0] = 1; s = "1"; a[n_] := a[n] = For[k = a[n-1] + 1, True, k++, If[StringFreeQ[s, t = ToString[k]], s = s <> t, Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 25 2013 *)

D=[]; for(n=1, 999, for(i=0, #D-#d=digits(n), D[i+1..i+#d]!=d || print1(n",") || next(2)); D=concat(D, d)) \\ M. F. Hasler, Oct 25 2019

Python
```
# see Hobson link
```

Edited by Patrick De Geest, Jun 02 2003

A131981 Number of early bird numbers <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 9, 9, 10, 10, 10, 10, 10, 10, 11, 12, 13, 14, 14, 15, 15, 15, 15, 15, 16, 17, 18, 19, 20, 20, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 27, 28, 28, 28, 29, 30, 31, 32

Offset: 1

Klaus Brockhaus, Aug 15 2007

a(n) = number of k such that A116700(k) <= n; a(n) = n - number of k such that A131881(k) <= n.

A131982 gives numbers n such that a(n) = n/2, or numbers n such that (number of k such that A116700(k) <= n) = (number of k such that A131881(k) <= n).

			There are two early bird numbers <= 21, viz. 12 and 21, hence a(21) = 2.

Klaus Brockhaus, Table of n, a(n) for n = 1..6000
Klaus Brockhaus, Plots of A131981(n)/n at various scales

Cf. A116700 (early bird numbers), A131881 (complement of A116700), A132133 (number of n-digit terms of 131881), A105390 (number of Rollman numbers <= n), A131982 (numbers n such that A131981(n) = n/2).

s$ = "" : d = 0
FOR n = 1 TO 84
sn$ = str$(n)
IF instr(s$, sn$) > 0 THEN d = d+1
s$ = s$ + sn$ : print d ; ",";
NEXT

A105391 Numbers m such that there are an equal number of numbers <= m that are contained and that are not contained in the concatenation of terms <= m in A048991.

Original entry on oeis.org

740, 1260, 1262, 5230, 15804, 15814, 15816, 36294, 194876, 213868

Offset: 1

Reinhard Zumkeller, Apr 04 2005

A105390(a(n)) = a(n)/2.

There are no other terms <= 600000. The plots in a105390.gif strongly suggest that the sequence is complete. - Klaus Brockhaus, Aug 15 2007

			A105390(n) < n/2 for n < a(1)=740;
A105390(n) > n/2 for n with 740 < n < a(2)=1260;
A105390(1261)=631, A105390(a(3))=A105390(1262)=631;
A105390(n) < n/2 for n with 1262 < n < a(4)=5230;
A105390(n) > n/2 for n with 5230 < n < a(5)=15804;
A105390(n) < n/2 for n with 15804 < n < a(6)=15814;
A105390(15815)=7908, A105390(a(7))=A105390(15816)=7909;
A105390(n) < n/2 for n with 15816 < n < a(8)=36294;
A105390(n) > n/2 for n with 36294 < n < a(9)=194876; etc.

Nick Hobson, Python program for this sequence

Cf. A048991, A048992, A105390, A131982 (numbers n such that A131981(n) = n/2).

s$ = "" : c = 0 : d = 0
FOR n = 1 TO 40000
sn$ = str$(n)
IF instr(s$, sn$) > 0 THEN d = d+1 ELSE c = c+1 : s$ = s$ + sn$
IF c = d THEN print n ; "," ;
NEXT ' Klaus Brockhaus, Aug 15 2007

A131982 Numbers n such that A131981(n) = n/2.

Original entry on oeis.org

576, 584, 588, 592, 600, 1650, 1654, 3430, 3440, 3448, 3452, 3458, 3462, 3466, 3474, 3520, 3600, 3608, 3610

Offset: 1

Klaus Brockhaus, Aug 15 2007

Numbers n such that number of terms <= n of A116700 equals number of terms <= n of A131881.
Numbers n such that numbers of numbers that occur in the concatenation of 1,2,3...,n-1 equals numbers of numbers that do not occur in the concatenation of 1,2,3...,n-1.
There are no other terms <= 600000. The plots in the link strongly suggest that the sequence is complete.

			A131981(n) < n/2 for 1 <=n < 576,
A131981(n) < n/2 for 576 < n < 584,
A131981(n) > n/2 for 584 < n < 588,
A131981(n) < n/2 for 588 < n < 592,
A131981(n) > n/2 for 592 < n < 600,
A131981(n) > n/2 for 600 < n < 1650,
A131981(n) > n/2 for 1650 < n < 1654,
A131981(n) < n/2 for 1654 < n < 3430,
A131981(n) > n/2 for 3430 < n < 3440,
..............
A131981(n) < n/2 for 3608 < n <= 3610,
A131981(n) > n/2 for 3610 < n <= 600000.

Klaus Brockhaus, Plots of A131981(n)/n at various scales

Cf. A116700 (early bird numbers), A131881 (complement of A116700), A131981 (number of early bird numbers <= n), A105390 (number of Rollman numbers <= n), A105391 (numbers n such that A105390(n) = n/2).

s$ = "" : c = 0 : d = 0
FOR n = 1 TO 4000
sn$ = str$(n)
IF instr(s$, sn$) > 0 THEN d = d+1 ELSE c = c+1
s$ = s$ + sn$ : IF c = d THEN print n ; ",";
NEXT

Edited by Charles R Greathouse IV, Oct 28 2009

cp's OEIS Frontend

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.

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A048992 Hannah Rollman's numbers: the numbers excluded from A048991.

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Haskell

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A131981 Number of early bird numbers <= n.

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JBASIC

A105391 Numbers m such that there are an equal number of numbers <= m that are contained and that are not contained in the concatenation of terms <= m in A048991.

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Programs

JBASIC

A131982 Numbers n such that A131981(n) = n/2.

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JBASIC

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