

A355497


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.


4



0, 4, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80
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OFFSET

1,2


COMMENTS

All 2digit numbers are terms.
All numbers having 0 as a digit (A011540) are terms, because p = 0, x^2  s*x + p = x*(xs) and the roots 0 and s are integers.


LINKS



FORMULA



EXAMPLE

k = 14 is a term, since the sum of the digits of 14 is 5, the product of the digits of 14 is 4 and the roots 1 and 4 of x^2  5x + 4 are all integers.


MATHEMATICA

kmax=80; kdig:=IntegerDigits[k]; s:=Total[kdig]; p:=Product[Part[kdig, i], {i, Length[kdig]}]; a:={}; For[k=0, k<=kmax, k++, If[Element[x/.Solve[x^2s*x+p==0, x], Integers], AppendTo[a, k]]]; a (* Stefano Spezia, Jul 06 2022 *)


PROG

(PARI) is(n)=my(v=if(n, digits(n), [0])); issquare(vecsum(v)^24*vecprod(v))


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



