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A217489
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Least positive integer without a digit 1, not listed earlier and not divisible by any digit of the preceding term.
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2
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2, 3, 4, 5, 6, 7, 8, 9, 20, 23, 25, 27, 29, 33, 22, 35, 26, 37, 32, 43, 34, 38, 28, 39, 40, 30, 44, 42, 45, 46, 47, 50, 24, 49, 53, 52, 57, 36, 55, 48, 54, 58, 59, 56, 62, 63, 64, 65, 67, 68, 69, 70, 60, 73, 74, 66, 75, 72, 79, 76, 80, 77, 78, 82, 83, 85, 84, 86, 87, 89, 92, 93, 88, 90
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OFFSET
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1,1
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COMMENTS
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This sequence contains all terms of A052383 that are not divisible by 2520. - Peter Kagey, Nov 04 2015
Here is a proof of Peter Kagey's comment:
Any number x in A052383 will eventually appear in the sequence if there are infinitely many members of the sequence containing no digit that divides x.
If k in A052383 is coprime to 210 (and thus not divisible by any digit > 1), then k is in the sequence.
The numbers 2...23 with number of 2's not divisible by 3, and 5...57 with number of 5's == 2,4 or 5 (mod 6) are coprime to 210, and thus are in the sequence.
The repunits k...k with k = 5 or 7 and an even number of digits are not divisible by 2 or 3, and thus they are in the sequence.
The repunits k...k with k = 2,3,4,6,8, or 9 and number of digits not divisible by 6 are not divisible by 5 or 7, and thus they are in the sequence. Any x in A052383 not divisible by 2520 is not divisible by one of the digits 2,3,...9, and thus is in the sequence. (End)
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LINKS
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MAPLE
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N:= 1000: # to get all terms before the first that exceeds N
A[1]:= 2:
Av:= remove(t -> has(convert(t, base, 10), 1), {$3..N}):
for n from 2 do
d:= convert(convert(A[n-1], base, 10), set) minus {0};
Ad:= remove(t -> ormap(y -> t mod y = 0, d) , Av);
if nops(Ad) = 0 then break fi;
A[n]:= min(Ad);
Av:= Av minus {A[n]};
od:
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MATHEMATICA
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a = {2}; Do[k = 1; While[Or[First@ DigitCount@ k > 0, MemberQ[a, k], Total[Boole@ Divisible[k, #] & /@ (IntegerDigits@ a[[n - 1]] /. 0 -> Nothing)] > 0], k++]; AppendTo[a, k], {n, 2, 74}]; a (* Michael De Vlieger, Nov 05 2015 *)
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PROG
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(PARI >= 2.6) A217489_vect(Nmax)={my(a=[], d=[0], u=0, nd); while( #a<Nmax, for( t=2, 9e9, bittest(u, t) & next; for(i=1+!d[1], #d, t%d[i] || next(2)); nd=Set(digits(t)); setsearch(nd, 1) & next; a=concat(a, t); u+=1<<t; d=nd; break)); a}
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CROSSREFS
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Sequence A217491 is a variant of the same idea (where injectivity is strengthened to strict monotonicity).
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KEYWORD
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AUTHOR
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STATUS
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approved
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