A355547 Numbers k such that x^2 - s*x + p has noninteger roots with s sum of digits of k and p product of digits of k.

1, 2, 3, 5, 6, 7, 8, 9, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 123, 124, 125, 126, 127, 128, 129, 131, 132, 133, 135, 136, 137, 138, 139, 141, 142, 144, 145, 147, 148, 149, 151, 152, 153, 154, 155, 156, 157, 159, 161, 162, 163, 165, 167, 168, 169, 171

Offset: 1

Author

Stefano Spezia and Jean-Marc Rebert, Jul 06 2022

Keywords

• nonn
• base

Examples

			k = 111 is a term, since the sum of the digits of 111 is 3, the product of the digits of 111 is 1 and the roots (3 - sqrt(5))/2 and (3 + sqrt(5))/2 of x^2 - 3*x + 1 are not integers.
		

Crossrefs

Complement of A355497.

Cf. A007953, A007954.

Subsequence of A052382.

Programs

Mathematica

kmax=171;kdig:=IntegerDigits[k]; s:=Total[kdig]; p:=Product[Part[kdig,i],{i,Length[kdig]}]; a:={};For[k=0,k<=kmax,k++,If[Not[Element[x/.Solve[x^2-s*x+p==0,x],Integers]],AppendTo[a,k]]]; a

Formula

a(n) ~ A052382(n) ~ n^k, where k = log(10)/log(9) = 1.04795.... - Charles R Greathouse IV, Jul 07 2022