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A325148
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Squares which can be expressed as the product of a number and its reversal.
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8
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0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 400, 484, 900, 1089, 1600, 1936, 2500, 3025, 3600, 4356, 4900, 5929, 6400, 7744, 8100, 9801, 10000, 10201, 12100, 12321, 14641, 17161, 19881, 22801, 25921, 29241, 32761, 36481, 40000, 40804, 44944, 48400, 49284, 53824, 58564, 63504, 68644, 73984, 79524, 85264
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listen;
history;
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OFFSET
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1,3
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COMMENTS
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The numbers k such that k * rev(k) is a square are in A306273.
The squares of palindromes of A014186 are a subsequence.
The square roots of the first 65 terms of this sequence (from 0 to 160000) are exactly the first 65 terms of A061917. Then a(66) = 162409 = 403^2 and the non-palindrome 403 is the first term of another sequence A325151.
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LINKS
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FORMULA
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EXAMPLE
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Zero ways: 169 = 13^2 cannot be equal to k * rev(k).
One way: 400 = 200 * 2; 10201 = 101 * 101; 162409 = 169 * 961.
Two ways: 7683984 = 2772 * 2772 = 1584 * 4851.
Three ways: 6350400 = 14400 * 441 = 25200 * 252 = 44100 * 144.
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MAPLE
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isA305231 := proc(n)
local d;
for d in numtheory[divisors](n) do
if d = digrev(n/d) then
return true ;
end if;
end do:
false ;
end proc:
n := 1;
for i from 0 to 4000 do
i2 := i^2 ;
if isA305231(i2) then
printf("%d %d\n", n, i2) ;
n := n+1 ;
end if;
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MATHEMATICA
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{0}~Join~Select[Range[10^3]^2, (d1=Select[Divisors[n=#], #<=Sqrt@n&]; Or@@Table[d1[[k]]==(IntegerReverse/@(n/d1))[[k]], {k, Length@d1}])&] (* Giorgos Kalogeropoulos, Jun 09 2021 *)
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PROG
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(Python)
from sympy import divisors
for n in range(10**6):
n2 = n**2
for m in divisors(n2):
if m > n:
break
if m == int(str(n2//m)[::-1]):
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CROSSREFS
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Subsequences: A014186 (square of palindromes), A076750 (product of a non-palindrome and its reversal, where leading zeros are not allowed).
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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