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A355608
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Zeroless numbers k such that x^2 - s*x + p has only integer roots, where s and p denote the sum and product of the digits of k respectively.
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1
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4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 122, 134, 143, 146
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OFFSET
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1,1
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COMMENTS
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There are respectively 1, 81, 52, 247, 650, 2335, 3129, 9100, 20682 terms with 1, 2, ..., 9 digits.
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LINKS
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EXAMPLE
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k = 4 is a term, since 4 is zeroless, the sum of the digits of 4 is 4, the product of the digits of 4 is 4 and the root 2 of x^2 - 4x + 4 is an integer.
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MAPLE
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isA355608 := proc(n)
local dgs, p, s ;
dgs := convert(n, base, 10) ;
p := mul(d, d=dgs) ;
s := add(d, d=dgs) ;
if p <> 0 then
-s/2+sqrt(s^2/4-p) ;
if type(simplify(%), integer) then
-s/2-sqrt(s^2/4-p) ;
if type(simplify(%), integer) then
true ;
else
false ;
end if;
else
false ;
end if;
else
false ;
end if ;
end proc:
for n from 1 to 180 do
if isA355608(n) then
printf("%d, ", n) ;
end if;
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PROG
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(PARI) is(n)=my(v=digits(n), c=vecprod(v)); c&& issquare(vecsum(v)^2-4*c)
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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