A299690 Numbers without digit 1 whose multiplicative digital root is not 0.

2, 3, 4, 5, 6, 7, 8, 9, 22, 23, 24, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 44, 46, 47, 48, 49, 53, 57, 62, 63, 64, 66, 67, 68, 72, 73, 74, 75, 76, 77, 79, 82, 83, 84, 86, 88, 89, 92, 93, 94, 97, 98, 99, 222, 223, 224, 226, 227, 228, 229, 232

Offset: 1

Author

J. Lowell, Feb 19 2018

Keywords

• nonn
• fini
• base

Comments

Is this sequence infinite?

There are no members of this sequence with 45 to 2000 decimal digits. Perhaps the last term is a(614640917006263790) = 77333222222222222222222222222222222222222222. - Charles R Greathouse IV, Feb 26 2018

This sequence is finite. Proof: Let k be the smallest term with more than 2000 decimal digits. Then the product of decimal digits pk of k has fewer than 2001 decimal digits (otherwise k isn't the smallest term with more than 2000 decimal digits). This number pk has at least as many decimal digits as 2^2001 has, which are 603. But then it doesn't have a nonzero multiplicative digital root per the computations of Charles R Greathouse IV. QED. - David A. Corneth, Aug 23 2018

Examples

			5 times 4 = 20 and 2 times 0 = 0, so 54 is not in this sequence.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A031347, A052383, A277061.

Programs

Mathematica

multDigRoot[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Select[Range[500], DigitCount[#, 10, 1] == 0 && multDigRoot[#] != 0 &] (* Alonso del Arte, Feb 19 2018, based on Robert G. Wilson v's program for A031347 *)

PARI

mdr(n)=while(n>9,n=factorback(digits(n)));n
do(n)=my(v=List());forvec(u=vector(n,i,[2,9]), if(mdr(factorback(u)), listput(v, fromdigits(u)))); Vec(v) \\ Gives n-digit elements
\\ Charles R Greathouse IV, Feb 19 2018

Formula

{ A052383 } intersect { A277061 }.