A355608 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.

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

Offset: 1

Author

Jean-Marc Rebert, Jul 09 2022

Keywords

• base
• nonn

Comments

Intersection of A052382 (zeroless numbers) and A355497.

There are respectively 1, 81, 52, 247, 650, 2335, 3129, 9100, 20682 terms with 1, 2, ..., 9 digits.

Examples

			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.
		

Links

• Jean-Marc Rebert, Table of n, a(n) for n = 1..3366

Crossrefs

Cf. A007953, A007954, A052382 (zeroless numbers).

Cf. A355497, A355547.

Programs

Maple

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;
end do: # R. J. Mathar, Jan 24 2023

PARI

is(n)=my(v=digits(n), c=vecprod(v)); c&& issquare(vecsum(v)^2-4*c)