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
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.
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s=[]; for(n=1, 100000, if(#vecsort(eval(Vec(Str(n^2))),,8)==9, s=concat(s, n^2))); s
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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
A294661
Numbers whose square contains all of the digits 1 through 9.
Original entry on oeis.org
11826, 12363, 12543, 14676, 15681, 15963, 18072, 19023, 19377, 19569, 19629, 20316, 22887, 23019, 23178, 23439, 24237, 24276, 24441, 24807, 25059, 25572, 25941, 26409, 26733, 27129, 27273, 29034, 29106, 30384, 32043, 32286, 33144, 34273, 35172, 35337, 35713, 35756, 35757, 35772, 35846, 35853
Offset: 1
11826^2 = 139854276 contains all digits from 1 to 9 exactly once.
The same is true for all terms up to 30384 whose square is 923187456. These terms are also listed in A071519, they form a subsequence of A054037.
The next 3 terms, 32043 (32043^2 = 1026753849), 32286 (32286^2 = 1042385796) and 33144 (33144^2 = 1098524736) contain all of the digits '0' through '9' exactly once: They are the first terms of A054038.
The next term, 34273 with 34273^2 = 1174638529, does not have this property, but the next two are again of that type (35172^2 = 1237069584 and 35337^2 = 1248703569).
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Select[Range[#, # + 3*10^4] &@ 11111, AllTrue[Most@ DigitCount[#^2], # > 0 &] &] (* Michael De Vlieger, Nov 08 2017 *)
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is_A294661(n)=#select(t->t,Set(digits(n^2)))>8
N=100;for(k=10^4,oo,is_A294661(k)||next;print1(k",");N--||break)
A204691
Numbers n such that n contains exactly 5 digits, all distinct, and n^2 contains exactly 9 distinct digits.
Original entry on oeis.org
10278, 12543, 12586, 13268, 13278, 13698, 14098, 15963, 16549, 16854, 17529, 18072, 19023, 20316, 20513, 20754, 21397, 21439, 23019, 23178, 24807, 25941, 26351, 26409, 27105, 27984, 28346, 28731, 29034, 29106
Offset: 1
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Select[Range[10000, Sqrt[10^9]], Length[Union[IntegerDigits[#]]] == 5 && Length[Union[IntegerDigits[#^2]]] == 9 &] (* T. D. Noe, Jan 18 2012 *)
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