A028820 -id:A028820 - cp's OEIS frontend

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

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

A028822 Squares with digits in nonincreasing order.

Original entry on oeis.org

0, 1, 4, 9, 64, 81, 100, 400, 441, 841, 900, 961, 6400, 7744, 8100, 10000, 40000, 44100, 84100, 90000, 96100, 640000, 774400, 810000, 1000000, 4000000, 4410000, 8410000, 8874441, 9000000, 9610000, 9853321, 64000000, 77440000

Offset: 1

Patrick De Geest

From Robert G. Wilson v, Jan 02 2014: (Start)
If x is present so is 100x. The primitives are 0, 1, 4, 9, 64, 81, 441, 841, 961, 7744, 8874441, 9853321, 999887641, …, . = A062826. Their square roots are: 0, 1, 2, 3, 8, 9, 21, 29, 31, 88, 2979, 3139, 31621, …, . Are there no more primitives?
Number of terms less than 10^k, beginning with k=0: 1, 4, 6, 12, 15, 21, 24, 32, 35, 44, 47, 56, 59, 68, 71, 80, 83, 92, 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 < 10^18: {0, 84}, {1, 8}, {4, 3}, {5, 0}, {6, 0}, {9, 1}. (End)

Robert G. Wilson v, Table of n, a(n) for n = 1..96
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A062826, A028820, A028821, A009996.

fQ[n_] := Max[ Differences[ IntegerDigits[ n]]] < 1; Select[ Range[0, 9000]^2, fQ] (* Robert G. Wilson v, Jan 02 2014 *)
Select[Range[0,10^4]^2,GreaterEqual@@IntegerDigits[#]&] (* Ray Chandler, Jan 05 2014 *)

isA009996(n)=n=digits(n); for(i=2,#n,if(n[i]>n[i-1],return(0))); 1
is(n)=issquare(n) && isA009996(n) \\ Charles R Greathouse IV, Jan 02 2014

For n > 1, a(n) = A062826(i) * 10^j for some i and j. - Charles R Greathouse IV, Jan 02 2014

a(n) = A028821(n)^2. - Ray Chandler, Jan 05 2014

Better name from Robert G. Wilson v, Jan 02 2014

A028819 Numbers whose square has its digits in nondecreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 12, 13, 15, 16, 17, 34, 35, 37, 38, 67, 83, 106, 107, 116, 117, 167, 183, 334, 335, 337, 367, 383, 587, 667, 1633, 1667, 3334, 3335, 3337, 3367, 3383, 3667, 4833, 6667, 16667, 33334, 33335, 33337, 33367, 33667, 36667, 66667

Offset: 1

Patrick De Geest

It appears that from a(53) onwards all terms have nondecreasing digits and has one of the following forms: 16..67, 3..34, 3..35, 3..37, 3..367, 3..36..67, 36..67 and 6..67 and all number of such forms are terms. - Chai Wah Wu, Dec 07 2015

Charles R Greathouse IV and Chai Wah Wu, Table of n, a(n) for n = 1..422 (n = 1..107 from Charles R Greathouse IV).
Patrick De Geest, Palindromic Squares in bases 2 to 17

Cf. A000290, A028820.

okQ[n_]:=And@@(#[[2]]>=#[[1]]&/@Partition[IntegerDigits[n^2],2,1])
Select[Range[0,50000],okQ]  (* Harvey P. Dale, Jan 09 2011 *)
Select[Range[0,10^5],LessEqual@@IntegerDigits[#^2]&] (* 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, isqrt
A028819_list = [0] + [int(isqrt(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

A234848 Triangular numbers with digits in nondecreasing order.

Original entry on oeis.org

0, 1, 3, 6, 15, 28, 36, 45, 55, 66, 78, 136, 378, 666, 1128, 1225, 1378, 2278, 2346, 2556, 5778, 12246, 13366, 22366, 22578, 35778, 111156, 222778, 223446, 333336, 1113778, 1222266, 1457778, 2235555, 3557778, 22227778, 22234446, 111116778, 156777778, 222446778

Offset: 1

Zak Seidov, Dec 31 2013

Beyond 222446778, all terms are k(k+1)/2 for k = 2s7, 6s7, or 6s8, where s stands for any number of 6's. - T. D. Noe, Jan 01 2014

T. D. Noe, Table of n, a(n) for n = 1..328 (terms less than 10^200)

Cf. A000217, A028820.

inOrder[nums_] := Min[Differences[nums]] >= 0; t = {}; Do[tri = n*(n+1)/2; If[inOrder[IntegerDigits[tri]], AppendTo[t, tri]], {n, 0, 10^5}]; t (* T. D. Noe, Dec 31 2013 *)
Select[Accumulate[Range[0,22000]],Min[Differences[IntegerDigits[#]]]>=0&] (* Harvey P. Dale, Apr 06 2023 *)

from itertools import chain, count, islice, combinations_with_replacement
from sympy import integer_nthroot
def A234848_gen(): # generator of terms
    return chain((0,), (n for n in (int(''.join(i)) for l in count(1) for i in combinations_with_replacement('123456789',l)) if integer_nthroot(8*n+1,2)[1]))
A234848_list = list(islice(A234848_gen(),50)) # Chai Wah Wu, May 22 2022

A352721 Perfect cubes whose decimal digits appear in nonincreasing order.

Original entry on oeis.org

0, 1, 8, 64, 1000, 8000, 64000, 1000000, 8000000, 64000000, 1000000000, 8000000000, 64000000000, 1000000000000, 8000000000000, 64000000000000, 1000000000000000, 8000000000000000, 64000000000000000, 1000000000000000000, 8000000000000000000, 64000000000000000000

Offset: 1

Antonio Roldán, Mar 30 2022

			64 is in the sequence because it is a perfect cube (64 = 4^3) whose digits appear in nonincreasing order.

Cf. A004647, A028820, A028864, A234848, A273045.

Intersection of A000578 and A009996.

Select[Range[0, 4*10^6]^3, Max@ Differences[IntegerDigits[#]] <= 0 &] (* Amiram Eldar, Mar 30 2022 *)

ok(n) = digits(n) == vecsort(digits(n),,4) && ispower(n,3)

a(n) = A004647(n-1)^3.

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

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A028822 Squares with digits in nonincreasing order.

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A028819 Numbers whose square has its digits in nondecreasing order.

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A234848 Triangular numbers with digits in nondecreasing order.

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A352721 Perfect cubes whose decimal digits appear in nonincreasing order.

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