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

1,2

Comments

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.

Links

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

Formula

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

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^2-s*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)^2-4*vecprod(v))

Crossrefs

Complement of A355547. A011540 is a subsequence.

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

Sequence in context: A346751 A292910 A292954 * A045855 A244216 A240580

Adjacent sequences: A355494 A355495 A355496 * A355498 A355499 A355500

Keyword

nonn,base

Author

Jean-Marc Rebert, Jul 04 2022

Status

approved