A062826 -id:A062826 - cp's OEIS frontend

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

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

A028868 Primes that when squared give numbers with digits in nonascending order.

Original entry on oeis.org

2, 3, 29, 31

Offset: 1

Patrick De Geest

No other solutions below 2*10^9 (probably finite). - Dec 15 1999
No other solutions below 10^20. - David A. Corneth, Oct 28 2023, recomputed after a remark from Max Alekseyev, David A. Corneth, Aug 20 2024
Primes p such that p^2 is in A062826. - Max Alekseyev, Aug 18 2024

			From _David A. Corneth_, Oct 28 2023: (Start)
31 is in the sequence as 31 is prime and 31^2 = 961 which has its digits in nonascending order.
2979 is not in the sequence even though 2979^2 = 8874441 does have digits in nonascending order but 2979 = 3^2 * 331 is not prime. (End)

David A. Corneth, PARI program

Cf. A028869, A028865, A028866, A062826.

is(n) = my(d = digits(n^2)); d == vecsort(d,,4) && isprime(n) \\ David A. Corneth, Oct 28 2023

PARI
```
\\ see link for a faster program
```

Name clarified by Jon E. Schoenfield, Oct 27 2023

A355063 Perfect powers whose digits are in nonincreasing order and do not include 0.

Original entry on oeis.org

1, 4, 8, 9, 32, 64, 81, 441, 841, 961, 7744, 7776, 8874441, 9853321, 999887641

Offset: 1

Jon E. Schoenfield, Jun 16 2022

a(16) > 10^45 if it exists. - Michael S. Branicky, Jun 19 2022

Cf. A001597, A009996, A028822, A062826.

perfectPowerQ[n_] := n==1||GCD @@ FactorInteger[n][[All, 2]] > 1; (* A001597 *) Select[Range[10^5], perfectPowerQ[#] && Max[Differences[d=IntegerDigits[#]]]<1 && Count[d,0]==0&] (* Stefano Spezia, Jul 01 2025 *)

isok(m) = if (ispower(m), my(d=digits(m)); vecmin(d) && (d == vecsort(d,,4))); \\ Michel Marcus, Jun 17 2022

from sympy import perfect_power as pp
from itertools import count, islice, combinations_with_replacement as mc
def agen():
    yield 1
    for d in count(1):
        nd = (int("".join(m)) for m in mc("987654321", d))
        yield from sorted(filter(pp, nd))
print(list(islice(agen(), 14))) # Michael S. Branicky, Jun 16 2022

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A028822 Squares with digits in nonincreasing order.

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A028868 Primes that when squared give numbers with digits in nonascending order.

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PARI

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A355063 Perfect powers whose digits are in nonincreasing order and do not include 0.

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Mathematica

PARI

Python