A028822 Squares with digits in nonincreasing order.

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

Author

Patrick De Geest

Keywords

• nonn
• base

Comments

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)

Links

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

Crossrefs

Cf. A062826, A028820, A028821, A009996.

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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