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A359346
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Reversible pandigital square numbers.
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2
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1234549876609, 9066789454321, 123452587690084, 123454387666009, 123454987660900, 123456987654400, 123458987664100, 123478988652100, 125688987432100, 146678985432100, 445678965432100, 480096785254321, 900666783454321, 906678945432100, 10223418547690084
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OFFSET
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1,1
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COMMENTS
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These are perfect squares containing each digit from 0 to 9 at least once and still remain square numbers (not necessarily of the same length) when reversing the digits.
In 1905, inspired by a question about all pandigital square numbers containing each digit from 0 to 9 exactly once (cf. A036745, A156977), the British mathematician Allan Cunningham (1842-1928) asked for reversible and palindromic pandigital square numbers. In his answer, he gives possible solutions, but actually not the least possible numbers he was asking for in his question.
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LINKS
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Allan Cunningham, Question 15789, The Educational Times, and Journal of the College of Preceptors 58 (1905), nr. 530 (June 1), p. 273; Solution 15789, Ibid., 59 (1906), nr. 537 (Jan. 1), p. 39.
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EXAMPLE
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Sequence starts with 1111103^2 = 1234549876609 <~> 9066789454321 = 3011111^2, which is the smallest possible such number.
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PROG
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(Python)
from math import isqrt
from itertools import count, islice
def c(n): return len(set(s:=str(n)))==10 and isqrt(r:=int(s[::-1]))**2==r
def agen(): yield from (k*k for k in count(10**6) if c(k*k))
(PARI) isok(k) = if (issquare(k), my(d=digits(k)); (#Set(d) == 10) && issquare(fromdigits(Vecrev(d))); ); \\ Michel Marcus, Dec 31 2022
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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