A028819 -id:A028819 - 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

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

A234841 Integers n such that digits in n and n^2 are 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, 116, 117, 167, 334, 335, 337, 367, 667, 1667, 3334, 3335, 3337, 3367, 3667, 6667, 16667, 33334, 33335, 33337, 33367, 33667, 36667, 66667, 166667, 333334, 333335, 333337, 333367, 333667, 336667

Offset: 1

Zak Seidov, Dec 31 2013

Zak Seidov, Table of n, a(n) for n = 1..96

Intersection of A009994 and A028819.

Select[Range[0,500000],LessEqual@@IntegerDigits[#]&&LessEqual@@IntegerDigits[#^2]&](* Ray Chandler, Jan 02 2014 *)

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

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A028820 Squares with digits in nondecreasing order.

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Author

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Comments

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Programs

Mathematica

PARI

Python

Formula

Extensions

A234841 Integers n such that digits in n and n^2 are in nondecreasing order.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica