A254143 -id:A254143 - cp's OEIS frontend

A254338 Initial digits of A254143 in decimal representation.

1, 4, 7, 1, 2, 3, 3, 4, 6, 1, 1, 2, 2, 2, 3, 3, 3, 4, 6, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Reinhard Zumkeller, Feb 27 2015

a(n) = A000030(A254143(n));
also initial digits of A254323: a(n) = A000030(A254323(n)).
all terms are of the form u*v mod 10, where u <= v and belonging to {1,3,4,6,7}, the distinct elements of A254397:
length of k-th run of consecutive 1s = A005993(k-2), k > 1;
length of k-th run of consecutive 2s = k*(k+1)/2 = A000217(k), k >= 1;
length of k-th run of consecutive 3s = k+1, k >= 1;
length of k-th run of consecutive 4s = A065033(k-1);
n with a(n) = 4: A237424(n) = (10^a+10^b+1)/3 with b = 0, see also A093137, A133384;
n with a(n) = 6: A237424(n) = (10^a+10^b+1)/3 with a = b; A005994(a(n)) = 6 for n > 1; see also A199682;

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

Cf. A254143, A254323, A000030, A254339, A005993, A000217, A065033.

Haskell
```
a254338 = a000030 . a254143
```

listA237424(lim)=my(v=List(),a,t); while(1, for(b=0,a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++)
do(lim)=my(v=List(),u=listA237424(lim),t); for(i=1,#u, for(j=1,i, t=u[i]*u[j]; if(t>lim,break); listput(v,t))); apply(n->digits(n)[1], Set(v)) \\ Charles R Greathouse IV, May 13 2015

A254339 Final digits of A254143 in decimal representation.

Original entry on oeis.org

1, 4, 7, 6, 8, 4, 7, 9, 7, 6, 8, 8, 9, 8, 4, 7, 7, 9, 7, 6, 8, 6, 8, 9, 8, 8, 8, 9, 9, 9, 8, 4, 7, 7, 7, 9, 9, 7, 6, 8, 8, 9, 8, 6, 8, 8, 9, 8, 8, 9, 8, 8, 9, 9, 9, 9, 9, 8, 4, 7, 7, 7, 7, 9, 9, 7, 6, 8, 6, 8, 9, 8, 8, 8, 9, 9, 9, 8, 6, 8, 8, 8, 9, 9, 8, 8

Offset: 1

Reinhard Zumkeller, Feb 23 2015

a(n) = A254143(n) mod 10;

also final digits of A254323: a(n) = A254323(n) mod 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to final digits of numbers

Cf. A254143, A254323, A010879, A254398, A254338.

Haskell
```
a254339 = flip mod 10 . a254143
```

listA237424(lim)=my(v=List(),a,t); while(1, for(b=0,a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++)
do(lim)=my(v=List(),u=listA237424(lim),t); for(i=1,#u, for(j=1,i, t=u[i]*u[j]; if(t>lim,break); listput(v,t))); apply(n->n%10, Set(v)) \\ Charles R Greathouse IV, May 13 2015

A254323 Remove in decimal representation of A254143(n) all repeated digits.

Original entry on oeis.org

1, 4, 7, 16, 28, 34, 37, 49, 67, 136, 148, 238, 259, 268, 34, 37, 367, 469, 67, 156, 1258, 136, 1348, 1369, 1468, 278, 238, 2359, 2479, 2569, 268, 34, 37, 367, 367, 489, 469, 67, 1356, 1458, 12358, 12469, 12478, 136, 1348, 13468, 13579, 1468, 2378, 2579

Offset: 1

Reinhard Zumkeller, Jan 28 2015

a(n) <= 123456789 for all n, and a(n) < 123456789 for n < 396;
a(396) = 123456789 = A050289(1);
a(n) = A137564(A254143(n)) = A004086(A227362(A254143(n))).

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

Cf. A254143, A137564, A227362, A004086, A050289.

Cf. A254338 (initial digits), A254339 (final digits).

Haskell
```
a254323 = a137564 . a254143
```

A009994 Numbers with digits in nondecreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122

Offset: 1

N. J. A. Sloane

Record values and occurrences of A004185. - Reinhard Zumkeller, Dec 05 2009
A193581(a(n)) = 0. - Reinhard Zumkeller, Aug 10 2011
This sequence was used by the U.S. Bureau of the Census in the mid-1950s to numerically code the alphabetical list of counties within a state (with some modifications for Texas). The 3-digit code has a "self-policing element" built into it and "was fairly effective in detecting the transposition of 2 digits." (Hanna 1959). - Randy A. Becker, Dec 11 2017

Amarnath Murthy and Robert J. Clarke, Some Properties of Staircase sequence, Mathematics & Informatics Quarterly, Volume 11, No. 4, November 2001.
Frank A. Hanna, The Compilation of Manufacturing Statistics. U.S. Department of Commerce, Bureau of the Census, 1959.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
David Applegate, Marc LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
David Radcliffe and Brendan McKay, Re: A009994, SeqFan list, Jul 29 2019
Eric Weisstein's World of Mathematics, Digit.
Index entries for 10-automatic sequences.

Apart from the first term, a subsequence of A052382. A254143 is a subsequence.

Cf. A152054, A036839.

import Data.Set (fromList, deleteFindMin, insert)
a009994 n = a009994_list !! n
a009994_list = 0 : f (fromList [1..9]) where
   f s = m : f (foldl (flip insert) s' $ map (10*m +) [m `mod` 10 ..9])
         where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 10 2011

[k:k in [0..122]|Sort(Intseq(k)) eq Reverse(Intseq(k))]; // Marius A. Burtea, Jul 28 2019

A[0]:= [0]: A[1]:= [$1..9]:
for d from 2 to 4 do
  A[d]:= map(t -> seq(10*t+i,i=(t mod 10) .. 9), A[d-1]):
od:
seq(op(A[d]),d=0..4); # Robert Israel, Jul 28 2019

Select[Range[0, 125], LessEqual@@IntegerDigits[#] &] (* Ray Chandler, Oct 25 2011 *)

is(n)=n=digits(n);n==vecsort(n) \\ Charles R Greathouse IV, Dec 03 2013

from itertools import combinations_with_replacement
def A009994generator():
    yield 0
    l = 1
    while True:
        for i in combinations_with_replacement('123456789',l):
            yield int(''.join(i))
        l += 1 # Chai Wah Wu, Nov 11 2015

def hasDigitsSorted(n: Int): Boolean = {
  val digSort = Integer.parseInt(n.toString.toCharArray.sorted.mkString)
  n == digSort
}
(0 to 200).filter(hasDigitsSorted()) // _Alonso del Arte, Apr 20 2020

a(n) >> exp(n^(1/10)). - Charles R Greathouse IV, Mar 15 2014

a(n) ~ 10^((9! n)^(1/9) - 5), since 10^(d - 1) <= a(n) < 10^d for binomial(d + 8, 9) < n <= binomial(d + 9, 9) = (d + 5 - epsilon)^9 / 9!. Using epsilon = 10/(3n) + o(1/n) gives more precise estimate. [Following Radcliffe and McKay, cf. SeqFan list.] - M. F. Hasler, Jul 30 2019

A237424 Numbers of the form (10^a + 10^b + 1)/3.

Original entry on oeis.org

1, 4, 7, 34, 37, 67, 334, 337, 367, 667, 3334, 3337, 3367, 3667, 6667, 33334, 33337, 33367, 33667, 36667, 66667, 333334, 333337, 333367, 333667, 336667, 366667, 666667, 3333334, 3333337, 3333367, 3333667, 3336667

Offset: 1

Ahmad J. Masad, Feb 07 2014

nonn

Has the property that the product of any two (not necessarily distinct) terms has digits in nondecreasing order.
Conjecture: This sequence is in a sense the maximally dense sequence with this nondecreasing products property. That is, it appears that every maximal sequence is either (i) A237424, (ii) a finite set of "extra" terms plus A237424 restricted to b=0 (which is A093137), or (iii) a finite set of "extra" terms plus A237424 restricted to a=b (which is A067275). (There might be one more case, not yet identified.) - David Applegate, Feb 12 2014
See A254143 and link for products a(i)*a(j) in natural order. - Reinhard Zumkeller, Jan 28 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1035 terms from Robert G. Wilson v)
Reinhard Zumkeller, First 10000 products of any two terms of A237424

Cf. A028819, A028820, A028821, A052216, A067275, A093137, A234841, A254143.

a237424 = flip div 3 . (+ 1) . a052216
-- Reinhard Zumkeller, Jan 28 2015

A052216:=[10^(n-1) + 10^(k-1): k in [1..n], n in [1..100]];
A237424:= func< n | (A052216[n]+1)/3 >;
[A237424(n): n in [1..100]]; // G. C. Greubel, Feb 22 2024

Union@ Flatten@ Table[(10^a + 10^b + 1)/3, {a, 0, 8}, {b, a, 8}] (* Robert G. Wilson v, Jan 26 2015 *)
(10^#[[1]]+10^#[[2]]+1)/3&/@Tuples[Range[0,8],2]//Union (* Harvey P. Dale, May 28 2019 *)

list(lim)=my(v=List(),a,t); while(1, for(b=0,a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++) \\ Charles R Greathouse IV, May 13 2015

from math import isqrt
def A237424(n): return (10**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+10**(n-1-(a*(a+1)>>1))+1)//3 # Chai Wah Wu, Apr 08 2025

A052216=flatten([[10^(n-1) + 10^(k-1) for k in range(1,n+1)] for n in range(1,101)])
def A237424(n): return (A052216[n-1]+1)//3
[A237424(n) for n in range(1,101)] # G. C. Greubel, Feb 22 2024

a(n) = (A052216(n) + 1)/3. - Reinhard Zumkeller, Jan 28 2015

Edited by David Applegate, Feb 07 2014

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A254338 Initial digits of A254143 in decimal representation.

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A254339 Final digits of A254143 in decimal representation.

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A254323 Remove in decimal representation of A254143(n) all repeated digits.

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A009994 Numbers with digits in nondecreasing order.

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A237424 Numbers of the form (10^a + 10^b + 1)/3.

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