A355547 -id:A355547 - cp's OEIS frontend

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.

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

Offset: 1

Jean-Marc Rebert, Jul 04 2022

All 2-digit numbers are terms.

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

			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.

Jean-Marc Rebert, Table of n, a(n) for n = 1..3002

Complement of A355547. A011540 is a subsequence.

Cf. A007953, A007954, A355574 (number of n-digit terms).

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^2-s*x+p==0,x],Integers],AppendTo[a,k]]]; a (* Stefano Spezia, Jul 06 2022 *)

is(n)=my(v=if(n,digits(n),[0])); issquare(vecsum(v)^2-4*vecprod(v))

a(n) = n + O(n^k) where k = log(9)/log(10) = 0.95424.... - Charles R Greathouse IV, Jul 07 2022

A355574 Number of nonnegative integers k with n digits 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.

Original entry on oeis.org

2, 90, 223, 2686, 31601, 370894, 4220160, 46962379, 512600193

Offset: 1

Stefano Spezia, Jul 07 2022

a(n) is the number of n-digit numbers in A355497.

Cf. A355497.

Cf. A007953, A007954, A063945, A355547.

a(n) <= A063945(n).

Limit_{n->oo} a(n)/a(n-1) = 10.

a(8)-a(9) from Jean-Marc Rebert, Jul 13 2022

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.

Original entry on oeis.org

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

Jean-Marc Rebert, Jul 09 2022

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.

			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.

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

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

Cf. A355497, A355547.

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

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

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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.

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A355574 Number of nonnegative integers k with n digits 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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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.

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