A237424 -id:A237424 - cp's OEIS frontend

A254143 Products of any two not necessarily distinct terms of A237424.

1, 4, 7, 16, 28, 34, 37, 49, 67, 136, 148, 238, 259, 268, 334, 337, 367, 469, 667, 1156, 1258, 1336, 1348, 1369, 1468, 2278, 2338, 2359, 2479, 2569, 2668, 3334, 3337, 3367, 3667, 4489, 4669, 6667, 11356, 11458, 12358, 12469, 12478, 13336, 13348, 13468, 13579

Offset: 1

Reinhard Zumkeller, Jan 28 2015

nonn

Digits are in nondecreasing order for all terms in decimal representation;
a(396) = 1123456789 = 3367 * 333667 is the smallest term containing all nonzero decimal digits: A254323(396) = 123456789;
A254323(n) = A137564(a(n)).

			Initial terms of A237424: 1, 4, 7, 34, 37, 67, 334, 337, 367, 667, 3334 ...
.  n | a(n) = A237424(i) * A237424(j)
. ---+-------------------------------
.  1 |    1 = 1 * 1   = A237424(1)^2
.  2 |    4 = 1 * 4   = A237424(1) * A237424(2)
.  3 |    7 = 1 * 7   = A237424(1) * A237424(3)
.  4 |   16 = 4 * 4   = A237424(2)^2
.  5 |   28 = 4 * 7   = A237424(2) * A237424(3)
.  6 |   34 = 1 * 34  = A237424(1) * A237424(4)
.  7 |   37 = 4 * 37  = A237424(1) * A237424(5)
.  8 |   49 = 7 * 7   = A237424(3)^2
.  9 |   67 = 1 * 67  = A237424(1) * A237424(6)
. 10 |  136 = 4 * 34  = A237424(2) * A237424(4)
. 11 |  148 = 4 * 37  = A237424(2) * A237424(5)
. 12 |  238 = 7 * 34  = A237424(3) * A237424(4)
. 13 |  259 = 7 * 37  = A237424(3) * A237424(5)
. 14 |  268 = 4 * 67  = A237424(2) * A237424(6)
. 15 |  334 = 1 * 334 = A237424(1) * A237424(7)
. 16 |  337 = 1 * 337 = A237424(1) * A237424(8)
. 17 |  367 = 1 * 367 = A237424(1) * A237424(9)
. 18 |  469 = 7 * 67  = A237424(3) * A237424(6)
. 19 |  667 = 1 * 34  = A237424(1) * A237424(10)
. 20 | 1156 = 34 * 34 = A237424(4)^2
see link for more.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, First 10000 products of any two terms of A237424

Subsequence of A009994.

Cf. A237424, A254323, A137564, A254338 (initial digits), A254339 (final digits).

import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null)
a254143 n = a254143_list !! (n-1)
a254143_list = f a237424_list [] empty where
   f xs'@(x:xs) zs s
     | Set.null s || x < y = f xs zs' (union s $ fromList $ map (* x) zs')
     | otherwise           = y : f xs' zs s'
     where zs' = x : zs
           (y, s') = deleteFindMin s

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++)
list(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))); Set(v) \\ Charles R Greathouse IV, May 13 2015

A254397 Initial digits of A237424 in decimal representation.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 23 2015

a(n) = A000030(A237424(n));
n with a(n) = 3: A237424(n) = (10^a+10^b+1)/3 with 0 <= b < a, see also A014132;
n with a(n) = 6: A237424(n) = (10^a+10^b+1)/3 with a = b, see also A199682, A254338;
length of k-th run of consecutive 3s = k+1, k > 0;
digits 0, 2, 5, 8 and 9 do not occur.

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

Cf. A237424, A000030, A254339.

Haskell
```
a254397 = a000030 . a237424
```

A254398 Final digits of A237424 in decimal representation.

Original entry on oeis.org

1, 4, 7, 4, 7, 7, 4, 7, 7, 7, 4, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Reinhard Zumkeller, Feb 23 2015

a(n) = A237424(n) mod 10;
n with a(n) = 4: A237424(n) = (10^a+10^b+1)/3 with b = 0, see also A093137, A133384;
n with a(n) = 7: A237424(n) = (10^a+10^b+1)/3 with 0 < b <= a;
length of k-th run of consecutive 7s = k;
digits 0, 2, 3, 5, 6, 8 and 9 do not occur.

Index entries for sequences related to final digits of numbers

Cf. A237424, A010879, A254339, A093137, A133384.

Haskell
```
a254398 = flip mod 10 . a237424
```

A052216 Sums of two powers of 10.

Original entry on oeis.org

2, 11, 20, 101, 110, 200, 1001, 1010, 1100, 2000, 10001, 10010, 10100, 11000, 20000, 100001, 100010, 100100, 101000, 110000, 200000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000001, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 20000000

Offset: 1

Henry Bottomley, Feb 01 2000

Numbers whose digit sum is 2.
A007953(a(n)) = 2; number of repdigits = #{2,11} = A242627(2) = 2. - Reinhard Zumkeller, Jul 17 2014
By extension, numbers k such that digitsum(k)^2 - 1 is prime. (PROOF: For any number k whose digit sum d > 2, d^2 - 1 = (d+1)*(d-1) and thus is not prime.) - Christian N. K. Anderson, Apr 22 2024

			From _Bruno Berselli_, Mar 07 2013: (Start)
The triangular array starts (see formula):
        2;
       11,      20;
      101,     110,     200;
     1001,    1010,    1100,    2000;
    10001,   10010,   10100,   11000,   20000;
   100001,  100010,  100100,  101000,  110000,  200000;
  1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000;
  ...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (terms 1..48 from Vincenzo Librandi, terms 49..1036 from T. D. Noe)

Subsequence of A069263 and A107679. A038444 is a subsequence.
Sums of n powers of 10: A011557 (1), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A002262, A003056, A007953, A242614, A242627, A237424.

a052216 n = a052216_list !! (n-1)
a052216_list = 2 : f [2] 9 where
   f xs@(x:_) z = ys ++ f ys (10 * z) where
                  ys = (x + z) : map (* 10) xs
-- Reinhard Zumkeller, Jan 28 2015, Jul 17 2014

[n: n in [1..10100000] | &+Intseq(n) eq 2]; // Vincenzo Librandi, Mar 07 2013

/* As a triangular array: */ [[10^n+10^m: m in [0..n]]: n in [0..8]]; // Bruno Berselli, Mar 07 2013

t = 10^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
With[{nn=7},Sort[Join[Table[FromDigits[PadRight[{2},n,0]],{n,nn}], FromDigits/@Flatten[Table[Table[Insert[PadRight[{1},n,0],1,i]],{n,nn},{i,2,n+1}],1]]]] (* Harvey P. Dale, Nov 15 2011 *)
Select[Range[10^9], Total[IntegerDigits[#]] == 2&] (* Vincenzo Librandi, Mar 07 2013 *)
T[n_,k_]:=10^(n-1)+10^(k-1); Table[T[n,k],{n,8},{k,n}]//Flatten (* Stefano Spezia, Nov 03 2023 *)

a(n)=my(d=(sqrtint(8*n)-1)\2,t=n-d*(d+1)/2-1); 10^d + 10^t \\ Charles R Greathouse IV, Dec 19 2016

from itertools import count, islice
def agen(): yield from (10**i + 10**j for i in count(0) for j in range(i+1))
print(list(islice(agen(), 34))) # Michael S. Branicky, May 15 2022

from math import isqrt
def A052216(n): return 10**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+10**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

def A052216(n,k): return 10^(n-1) + 10^(k-1)
flatten([[A052216(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Feb 22 2024

T(n,k) = 10^(n-1) + 10^(k-1) with 1 <= k <= n.
a(n) = 3*A237424(n) - 1. - Reinhard Zumkeller, Jan 28 2015
a(n) = 10^A003056(n-1) + 10^A002262(n-1). - Chai Wah Wu, Apr 08 2025

A254338 Initial digits of A254143 in decimal representation.

Original entry on oeis.org

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

A028820 Squares with digits in nondecreasing order.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 144, 169, 225, 256, 289, 1156, 1225, 1369, 1444, 4489, 6889, 11236, 11449, 13456, 13689, 27889, 33489, 111556, 112225, 113569, 134689, 146689, 344569, 444889, 2666689, 2778889, 11115556, 11122225, 11135569

Offset: 1

Patrick De Geest

Number of terms less than 10^k, beginning with k=0: 1, 4, 8, 13, 19, 25, 32, 34, 42, 43, 50, 53, 61, 62, 71, 72, 82, 83, 94, 95, …, .

Like all squares the ending digits can be 0, 1, 4, 5, 6 or 9. Here is the tally of the list of terms < 10^19: {0, 1}, {1, 1}, {4, 4}, {5, 10}, {6, 13}, {9, 66}. Robert G. Wilson v, Jan 01 2014

Robert G. Wilson v and Chai Wah Wu, Table of n, a(n) for n = 1..428 (n = 1..106 from Robert G. Wilson v).
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A028819, A028821, A028822, A237424.

Intersection of A000290 and A009994.

Select[Range[0,4000]^2,Min[Differences[IntegerDigits[#]]]>-1&] (* Harvey P. Dale, Dec 31 2013 *)
Select[Range[0,10^4]^2,LessEqual@@IntegerDigits[#]&] (* Ray Chandler, Jan 06 2014 *)

mono(n)=n=eval(Vec(Str(n)));for(i=2,#n,if(n[i]Charles R Greathouse IV, Aug 22 2011

from itertools import combinations_with_replacement
from gmpy2 import is_square
A028820_list = [0] + [n for n in (int(''.join(i)) for l in range(1,11) for i in combinations_with_replacement('123456789',l)) if is_square(n)] # Chai Wah Wu, Dec 07 2015

a(n) = A028819(n)^2. - Ray Chandler, Jan 06 2014

Definition edited by Zak Seidov, Dec 31 2013

A374258 Square array: T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3, read by descending antidiagonals.

Original entry on oeis.org

1, 4, 1, 34, 10, 1, 334, 262, 28, 1, 3334, 7318, 2242, 82, 1, 33334, 204886, 183790, 19846, 244, 1, 333334, 5736790, 15070726, 4842262, 177634, 730, 1, 3333334, 160630102, 1235799478, 1181511766, 129672334, 1595782, 2188, 1, 33333334, 4497642838, 101335557142, 288288870742, 94660803334, 3491569558, 14353282, 6562, 1

Offset: 1

Ahmad J. Masad, Jul 01 2024

This sequence gives the matrix M in the definition of A365450. Similar to A266577.
Conjecture: For each natural number n, the digits of the product of any (n+1) not necessarily distinct terms of the n-th row in the base (3^(n+1)+1) numeral system appear in nondecreasing order.
Proof of the conjecture. Let b := 3^(n+1)+1. The product of any (n+1) terms of the n-th row has the form p/(b-1), where p is the product of (n+1) numbers of the form b^k+2. Let p = (dm, ..., d1, d0)b, and we have d0+d1+...+dm = 3^(n+1) = b-1. Then p/(b-1) = (dm, ..., d2+...+dm, 1+d1+...+dm)_b, which do form a nondecreasing sequence. - _Max Alekseyev, Jul 03 2024
The preceding result is similar to the property of the nondecreasing products mentioned in A237424. Specifically; the first row of this array is A093137, which is a subsequence of A237424.  - Ahmad J. Masad, Jul 30 2024
More generally: Let r and n be positive integers and S be a sequence of all numbers of the form (b^c(1)+b^c(2)+...+b^c(r)+1)/(r+1), where c(1),...,c(r) are nonnegative integers. Then in the numeral system base b := (r+1)^(n+1)+1, the digits of the product of any n+1 (possibly equal) terms of S appear in nondecreasing order. Proof is similar. - Ahmad J. Masad, Jul 30 2024; edited by Max Alekseyev, Aug 01 2024

			The array begins:
  1    4    34   334   3334
  1   10   262  7318
  1   28  2242
  1   82
  1
Example of the conjecture: Take 5 terms from the 4th row and find their product in base 244 numeral system (since 3^(4+1)+1=244) as follows: 82,19846 twice and 4842262 twice, the product is equal to 82*19846*19846*4842262*4842262 = 757279838666167487626528 = (1, 3, 7, 19, 31, 55, 91, 115, 163, 195, 212)_244 which is in agreement with the conjecture since the digits in 244 base numeral system are in nondecreasing order.
Example of the general property: Take r=3 and n=4, then b=4^5+1=1025. The sequence S is the sequence of the numbers of the form (1025^b(1)+1025^b(2)+1025^b(3)+1)/4. Let's multiply 5 terms of the sequence S, say ((1025^0+1025^0+1025^1+1)/4)*(((1025^0+1025^1+1025^1+1)/4)^2)*(((1025^3+1025^4+1025^4+1)/4)^2) = 257*513^2*552175667969^2 = 20621601208620337073958261562113 = (16,112,308,488,580,680,832,936,964,984,1013)_1025. The digits of the product in base 1025 are in nondecreasing order.

Michel Marcus, Table of n, a(n) for n = 1..820

Cf. A093137, A266577, A365450, A237424.

T[n_, k_] := ((3^(n+1) + 1)^(k-1) + 2)/3; Table[T[k, n-k+1], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 02 2024 *)

T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3 \\ Andrew Howroyd, Jul 01 2024

A384094 Numbers whose square has digit sum 9 and no trailing zero.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 39, 45, 48, 51, 102, 105, 111, 201, 249, 318, 321, 348, 351, 501, 549, 1002, 1005, 1011, 1101, 1149, 1761, 2001, 4899, 5001, 10002, 10005, 10011, 10101, 10149, 11001, 14499, 20001, 50001, 100002, 100005, 100011, 100101, 101001, 110001, 200001, 375501, 500001, 1000002

Offset: 1

M. F. Hasler, Jun 15 2025

All numbers of the form 10^a + 10^b + 1 (i.e., A052216+1 = 3*A237424) and of the form 10^a + 5*10^b with min(a, b) = 0 (i.e., A133472 U A199685), are in this sequence. Terms not of this form are (9, 18, 39, 45, 48, 249, 318, 321, 348, 351, 549, 1149, 1761, 4899, 10149, 14499, 375501, ...), see subsequence A384095. (Is this sequence finite? What is the next term?)

Is it true that no number > 1049 = A215614(6) has a square with digit sum less than 9, other than the trivial 1 and 4?

Cf. A004159 (sum of digits of n^2), A215614 (sumdigits(n^2) = 7), A133472 (10^n + 5), A199685 (5*10^n + 1), A052216 (10^a + 10^b), A237424 ((10^a + 10^b + 1)/3).

See also: A058414 (digits(n^2) in {0,1,4}).

select( {is_A384094(n)=n%10 && sumdigits(n^2)==9}, [1..10^5])

A384095 Numbers other than {10^a + 10^b + 1} and {10^a + 5*10^b, min(a, b) = 0} whose square has digit sum 9 and no trailing zero.

Original entry on oeis.org

9, 18, 39, 45, 48, 249, 318, 321, 348, 351, 549, 1149, 1761, 4899, 10149, 14499, 375501

Offset: 1

M. F. Hasler, Jun 15 2025

The definition excludes the two "regular" subsequences of A384094, namely A052216+1 = 3*A237424 and A133472 U A199685, which provide most of its terms.
Is it true that no number > 1049 = A215614(6) has a square with digit sum less than 9, other than the trivial 1 and 4?
The next term, if it exists, is a(18) > 10^8.
a(18) > 10^14 if it exists. - Robert Israel, Jun 15 2025
a(18) > 10^40 if it exists. - Chai Wah Wu, Jun 19 2025

Cf. A004159 (sum of digits of n^2), A384094 (sumdigits(n^2) = 9), A133472 (10^n+5), A199685 (5*10^n + 1), A052216 (10^a+10^b), A237424 ((10^a+10^b+1)/3).

See also: A215614 (sumdigits(n^2) = 7), A058414 (digits(n²) ⊂ {0,1,4}).

extend:= proc(a,d) local i,s;
    s:= convert(convert(a,base,10),`+`);
    op(select(t -> numtheory:-quadres(t,10^d)=1, [seq(i*10^(d-1)+a, i=0 .. 9 - s)]))
end proc:
istriv:= proc(n) local L;
   L:= subs(0=NULL,convert(n,base,10));
   member(L, [[4],[5],[6],[1,1],[1,1,1],[1,2],[2,1],[1,5],[5,1]])
end proc:
R:= NULL:
A:= [1,4,5,6,9]:
for d from 2 to 20 do
  A:= map(extend,A,d);
  V:= select(t -> t > 10^(d-1) and issqr(t) and convert(convert(t,base,10),`+`)=9, A);
  if V <> [] then V:= sort(remove(istriv,map(sqrt,V))); R:= R,op(V); fi
od:
R;# Robert Israel, Jun 15 2025

select( {is_A384095(n)=n%10 && sumdigits(n^2)==9 && !bittest(36938, fromdigits(Set(digits(n))))}, [1..10^5])

A284691 Numbers of the form (10^c-1)*the product any two (not necessarily distinct) terms of A074992.

Original entry on oeis.org

9, 99, 333, 999, 3663, 9999, 12321, 30303, 36963, 99999, 135531, 333333, 369963, 999999, 1121211, 1367631, 3003003, 3363633, 3699963, 9999999, 12333321, 13688631, 33033033, 33666633, 36999963, 99999999, 102030201, 111111111, 124454421, 136898631, 300030003

Offset: 1

Ahmad J. Masad, Apr 01 2017

Conjecture 1: all terms are palindromic in base 10.

Conjecture 2: the sequence A074992 is the maximally dense sequence with this palindromic products property.

			a(3) = 37*9 = 333, with respect to strictly increasing ordering.

Cf. A002283, A074992, A237424.

f[n_] := f[n] = (10^(2 n) + 10^n + 1)/3; c[n_] := 10^n - 1; mx = 10^10; i=1; Union@ Reap[ While[c[i] <= mx, j=0; While[c[i] f[j] <= mx, k=0; While[k <= j && (v = c[i] f[j] f[k]) <= mx, Sow@v; k++]; j++]; i++]][[2, 1]] (* Giovanni Resta, Apr 01 2017 *)

a(13)-a(31) from Giovanni Resta, Apr 01 2017

cp's OEIS Frontend

A254143 Products of any two not necessarily distinct terms of A237424.

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A254397 Initial digits of A237424 in decimal representation.

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A254398 Final digits of A237424 in decimal representation.

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A052216 Sums of two powers of 10.

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A254338 Initial digits of A254143 in decimal representation.

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A028820 Squares with digits in nondecreasing order.

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A374258 Square array: T(n,k) = ((3^(n+1) + 1)^(k-1) + 2)/3, read by descending antidiagonals.

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