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A299690
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Numbers without digit 1 whose multiplicative digital root is not 0.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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5 times 4 = 20 and 2 times 0 = 0, so 54 is not in this sequence.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,fini,base
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AUTHOR
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STATUS
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approved
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