A225218 -id:A225218 - cp's OEIS frontend

A235717 Squares which have one or more occurrences of exactly two different digits.

16, 25, 36, 49, 64, 81, 100, 121, 144, 225, 400, 441, 484, 676, 900, 1444, 7744, 10000, 11881, 29929, 40000, 44944, 55225, 69696, 90000, 1000000, 4000000, 9000000, 9696996, 100000000, 400000000, 900000000, 6661661161, 10000000000, 40000000000, 90000000000

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 100.

This sequence is the same as A018885, except that A018885 has four additional leading terms.

			69696 is in the sequence because 69696 = 264^2 and 69696 contains exactly two different digits: 6 and 9.

Cf. A235718-A235724, A225218.

s=[]; for(n=1,10000, if(#vecsort(eval(Vec(Str(n^2))),,8)==2, s=concat(s, n^2))); s

a(n) = A016069(n)^2.

A235718 Squares which have one or more occurrences of exactly three different digits.

Original entry on oeis.org

169, 196, 256, 289, 324, 361, 529, 576, 625, 729, 784, 841, 961, 1156, 1225, 1521, 1600, 1681, 2025, 2116, 2209, 2500, 3136, 3364, 3600, 3844, 3969, 4225, 4489, 4624, 4900, 5625, 5776, 5929, 6400, 6561, 6889, 7225, 8100, 8281, 8464, 8836, 9409, 10201, 10404

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 1156.

			5929 is in the sequence because 5929 = 77^2 and 5929 contains exactly three different digits: 2, 5 and 9.

Giovanni Resta, Table of n, a(n) for n = 1..12299 (terms < 10^36, first 600 terms from Colin Barker)

Cf. A054031, A225218, A235717, A235719-A235724.

s=[]; for(n=1, 200, if(#vecsort(eval(Vec(Str(n^2))),,8)==3, s=concat(s, n^2))); s

a(n) = A054031(n)^2.

A235724 Squares which have one or more occurrences of exactly nine different digits.

Original entry on oeis.org

102495376, 102576384, 102738496, 104325796, 105637284, 139854276, 152843769, 157326849, 158306724, 158407396, 172843609, 176039824, 176305284, 178035649, 180472356, 183467025, 187635204, 198753604, 208571364, 215384976, 217356049, 218034756, 235714609

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 1005397264.

The smallest penholodigital square is a(6) = A036744(1) = 139854276 and the largest one is a(83) = A036744(30) = 923187456 (see Penguin references). - Bernard Schott, Feb 07 2022

			102495376 is in the sequence because 102495376 = 10124^2 and 102495376 contains exactly nine different digits: 0, 1, 2, 3, 4, 5, 6, 7 and 9.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 139854276, page 184 and entry 923187456, page 186.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Colin Barker)

Cf. A054037.
Cf. A235717-A235723, A225218.
A036744 is a subsequence.

s=[]; for(n=1, 100000, if(#vecsort(eval(Vec(Str(n^2))),,8)==9, s=concat(s, n^2))); s

from itertools import count, islice
def agen(): yield from (r*r for r in count(10**4) if len(set(str(r*r)))==9)
print(list(islice(agen(), 23))) # Michael S. Branicky, May 24 2022

a(n) = A054037(n)^2.

A235723 Squares which have one or more occurrences of exactly eight different digits.

Original entry on oeis.org

10278436, 10673289, 10679824, 10837264, 13498276, 13527684, 13675204, 13860729, 13942756, 16378209, 16785409, 17280649, 17430625, 19847025, 20584369, 20738916, 21307456, 21473956, 21743569, 23078416, 23174596, 23970816, 24137569, 24671089, 24870169, 28901376

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 101465329.

			10278436 is in the sequence because 10278436 = 3206^2 and 10278436 contains exactly eight different digits: 0, 1, 2, 3, 4, 6, 7 and 8.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A235717-A235722, A235724, A225218.

s=[]; for(n=1, 10000, if(#vecsort(eval(Vec(Str(n^2))),,8)==8, s=concat(s, n^2))); s

a(n) = A054036(n)^2.

A235719 Squares which have one or more occurrences of exactly four different digits.

Original entry on oeis.org

1024, 1089, 1296, 1369, 1764, 1849, 1936, 2304, 2401, 2601, 2704, 2809, 2916, 3025, 3249, 3481, 3721, 4096, 4356, 4761, 5041, 5184, 5329, 5476, 6084, 6241, 6724, 7056, 7396, 7569, 7921, 8649, 9025, 9216, 9604, 9801, 10609, 10816, 11025, 11236, 12544, 12996

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 10609.

			5329 is in the sequence because 5329 = 73^2 and 5329 contains exactly four different digits: 2, 3, 5 and 9.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A235717, A235718, A235720-A235724, A225218.

Select[Range[150]^2,Length[Union[IntegerDigits[#]]]==4&] (* Harvey P. Dale, May 03 2018 *)

s=[]; for(n=1, 300, if(#vecsort(eval(Vec(Str(n^2))),,8)==4, s=concat(s, n^2))); s

a(n) = A054032(n)^2.

A235720 Squares which have one or more occurrences of exactly five different digits.

Original entry on oeis.org

12769, 13456, 13689, 13924, 15376, 15876, 16384, 17689, 17956, 18496, 18769, 20164, 20736, 21609, 21904, 23104, 23409, 23716, 28561, 29584, 30276, 30625, 30976, 31684, 32041, 32761, 34596, 35721, 36481, 37249, 38025, 38416, 39204, 39601, 41209, 43681, 45369

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 100489.

			30976 is in the sequence because 30976 = 176^2 and 30976 contains exactly five different digits: 0, 3, 6, 7 and 9.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A235717-A235719, A235721-A235724, A225218.

s=[]; for(n=1, 1200, if(#vecsort(eval(Vec(Str(n^2))),,8)==5, s=concat(s, n^2))); s

a(n) = A054033(n)^2.

A235721 Squares which have one or more occurrences of exactly six different digits.

Original entry on oeis.org

103684, 104329, 104976, 107584, 123904, 124609, 132496, 134689, 139876, 140625, 157609, 162409, 164025, 170569, 173056, 180625, 195364, 198025, 207936, 209764, 214369, 237169, 254016, 257049, 258064, 259081, 279841, 293764, 310249, 318096, 321489, 326041

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 1028196.

			124609 is in the sequence because 124609 = 353^2 and 124609 contains exactly six different digits: 0, 1, 2, 4, 6 and 9.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A235717-A235720, A235722-A235724, A225218.

s=[]; for(n=1, 1200, if(#vecsort(eval(Vec(Str(n^2))),,8)==6, s=concat(s, n^2))); s

a(n) = A054034(n)^2.

A235722 Squares which have one or more occurrences of exactly seven different digits.

Original entry on oeis.org

1034289, 1046529, 1048576, 1054729, 1056784, 1073296, 1075369, 1085764, 1238769, 1247689, 1354896, 1380625, 1382976, 1432809, 1507984, 1605289, 1607824, 1630729, 1695204, 1708249, 1750329, 1763584, 1803649, 1827904, 1836025, 1890625, 1946025, 1974025

Offset: 1

Colin Barker, Jan 15 2014

The first term having a repeated digit is 10137856.

			1247689 is in the sequence because 1247689 = 1117^2 and 1247689 contains exactly seven different digits: 1, 2, 4, 6, 7, 8 and 9.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A235717-A235721, A235723, A235724, A225218.

sddQ[n_]:=Count[DigitCount[n],?(#>0&)]==7; Select[Range[1001,1450]^2, sddQ] (* _Harvey P. Dale, Mar 12 2015 *)

s=[]; for(n=1, 10000, if(#vecsort(eval(Vec(Str(n^2))),,8)==7, s=concat(s, n^2))); s

a(n) = A054035(n)^2.

A339693 All pandigital squares which contain each digit exactly once in some base b >= 2. The numbers are written in base 10.

Original entry on oeis.org

225, 38025, 314721, 622521, 751689, 3111696, 6002500, 7568001, 10323369, 61058596, 73513476, 74545956, 94517284, 105144516, 112572100, 112656996, 132756484, 136936804, 181980100, 202948516, 210308004, 211353444, 219573124, 222069604, 230614596, 238208356, 251983876

Offset: 1

David Schilling, Dec 13 2020

The sequence consists of all square numbers which when represented in some base b contain all the b digits in that base exactly once.

A225218 has all the squares in base 10 that are pandigital. This sequence is the union of all such sequences in any integer base b >= 2.

			15^2 in base 4 (225 is 3201 in base 4) contains the digits 0-3.
195^2 in base 6 (38025 is 452013 in base 6) contains the digits 0-5.
The next three terms contain all the digits in base 7.
The following four entries are pandigital in base 8, the next 26 in base 9, and so on.

David Schilling, Table of n, a(n) for n = 1..5508

Cf. A036745, A134741, A258103, A260182, A340501.

#import "Basic";
dstr := "0123456789abcdef";
main :: () {
    digits : [16] int;
    for j:2..3_000_000 {
        for b:3..16 {
            for d : 0..15
                digits[d] = 0;
            k := j*j;
            s := tprint( "%",  formatInt( k, b ) );
            if s.count > b
                continue;
            for d : 0..s.count-1 {
                for c : 0..dstr.count-1 {
                    if s[d] == dstr[c] {
                        digits[c] += 1;
                        continue d;
                    }
                }
            }
            for d : 0..b-1 {
                if digits[d] != 1
                    continue b;
            }
            print( "%, ", k );
        }
    }
}

\\ here ispandig(n) returns base if n is pandigital, otherwise 0.
ispandig(n)={for(b=2, oo, my(r=logint(n,b)+1); if(rAndrew Howroyd, Dec 20 2020

A359342 Least pandigital square with n digits.

Original entry on oeis.org

1026753849, 10057482369, 100549873216, 1000574082369, 10000938205476, 100005740082369, 1000000973875264, 10000057400082369, 100000030347218596, 1000000574000082369, 10000000365759287524, 100000005740000082369, 1000000003751486308921, 10000000057400000082369

Offset: 10

Martin Renner, Dec 27 2022

Pandigital squares are perfect squares containing each digit from 0 to 9 at least once.

For number of digits n >= 11, every second term is of the form 10...05740...082369 with (n-1)/2 - 3 zeros after the leading 1 and (n-1)/2 - 5 zeros after the middle three digits 547. This term is 10...0287^2 with (n-1)/2 - 3 zeros after the leading 1. This is the case since (10^m + 287)^2 = 10^(2*m) + 574*10^m + 82369 with m = (n-1)/2 and n >= 11 odd, and is the first n-digit square containing all digits from 0 to 9.

Cf. A225218, A359343, A359344.

a:=proc(n::posint) local s, k, K: if n<10 then s:=NULL: else for k from ceil(sqrt(10^(n-1))) to floor(sqrt(10^n)) do K:=convert(k^2,base,10); if nops({op(K)})=10 then s:=k^2: break: fi: od: fi: return s; end:
seq(a(n),n=10..30);

from math import isqrt
def c(n): return len(set(str(n))) == 10
def a(n): return next((k*k for k in range(isqrt(10**(n-1))+1, isqrt(10**n-1)+1) if c(k*k)), None)
print([a(n) for n in range(10, 24)]) # Michael S. Branicky, Dec 27 2022

a(n) = 10^(n-1) + 574*10^((n-1)/2) + 82369 for n >= 11 odd.

cp's OEIS Frontend

A235717 Squares which have one or more occurrences of exactly two different digits.

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A235718 Squares which have one or more occurrences of exactly three different digits.

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A235724 Squares which have one or more occurrences of exactly nine different digits.

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A235723 Squares which have one or more occurrences of exactly eight different digits.

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A235719 Squares which have one or more occurrences of exactly four different digits.

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A235720 Squares which have one or more occurrences of exactly five different digits.

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A235721 Squares which have one or more occurrences of exactly six different digits.

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