%I #46 Jan 24 2023 10:30:26
%S 4,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,31,32,33,34,
%T 35,36,37,38,39,41,42,43,44,45,46,47,48,49,51,52,53,54,55,56,57,58,59,
%U 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
%N 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.
%C Intersection of A052382 (zeroless numbers) and A355497.
%C There are respectively 1, 81, 52, 247, 650, 2335, 3129, 9100, 20682 terms with 1, 2, ..., 9 digits.
%H Jean-Marc Rebert, <a href="/A355608/b355608.txt">Table of n, a(n) for n = 1..3366</a>
%e 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.
%p isA355608 := proc(n)
%p local dgs,p,s ;
%p dgs := convert(n,base,10) ;
%p p := mul(d,d=dgs) ;
%p s := add(d,d=dgs) ;
%p if p <> 0 then
%p -s/2+sqrt(s^2/4-p) ;
%p if type(simplify(%),integer) then
%p -s/2-sqrt(s^2/4-p) ;
%p if type(simplify(%),integer) then
%p true ;
%p else
%p false ;
%p end if;
%p else
%p false ;
%p end if;
%p else
%p false ;
%p end if ;
%p end proc:
%p for n from 1 to 180 do
%p if isA355608(n) then
%p printf("%d,",n) ;
%p end if;
%p end do: # _R. J. Mathar_, Jan 24 2023
%o (PARI) is(n)=my(v=digits(n), c=vecprod(v)); c&& issquare(vecsum(v)^2-4*c)
%Y Cf. A007953, A007954, A052382 (zeroless numbers).
%Y Cf. A355497, A355547.
%K base,nonn
%O 1,1
%A _Jean-Marc Rebert_, Jul 09 2022