A067581
a(n) = smallest integer not yet in the sequence with no digits in common with a(n-1), a(0)=0.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 11, 20, 13, 24, 15, 23, 14, 25, 16, 27, 18, 26, 17, 28, 19, 30, 12, 33, 21, 34, 29, 31, 40, 32, 41, 35, 42, 36, 44, 37, 45, 38, 46, 39, 47, 50, 43, 51, 48, 52, 49, 53, 60, 54, 61, 55, 62, 57, 63, 58, 64, 59, 66, 70, 56, 71, 65, 72, 68, 73, 69
Offset: 0
Ulrich Schimke (ulrschimke(AT)aol.com)
a(14) = 13, since a(13) = 20 and all integers smaller than 13 have a digit in common with 20 or have already appeared in the sequence.
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import Data.List (delete, intersect); import Data.Function (on)
a067581 n = a067581_list !! (n-1)
a067581_list = 1 : f 1 [2..] where
f u vs = v : f v (delete v vs)
where v : _ = filter (null . (intersect `on` show) u) vs
-- Reinhard Zumkeller, Jul 01 2013
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f[s_List] := Block[{k = 1, id = IntegerDigits@ s[[ -1]]}, While[ MemberQ[s, k] || Intersection[id, IntegerDigits@k] != {}, k++ ]; Append[s, k]]; Nest[f, {1}, 71] (* Robert G. Wilson v, Apr 03 2009 *)
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{u=0; a=0; for(n=0, 99, print1(a", "); u+=1<M. F. Hasler, Nov 01 2014
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from itertools import count, islice, product as P
def only(s, D=1): # numbers with >= D digits only from s
yield from (int("".join(p)) for d in count(D) for p in P(s, repeat=d))
def agen(): # generator of terms
aset, an, minan = {0}, 0, 1
while True:
yield an
an, s = minan, set(str(an))
use = "".join(c for c in "0123456789" if c not in s)
for an in only(use, D=len(str(minan))):
if an not in aset: break
aset.add(an)
while minan in aset: minan += 1
print(list(islice(agen(), 73))) # Michael S. Branicky, Jun 30 2022
A276633
a(n) = smallest integer not yet in the sequence with no digits in common with a(n-1) and a(n-2); a(0)=0, a(1)=1.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 33, 11, 20, 34, 15, 26, 30, 14, 25, 36, 17, 24, 35, 16, 27, 38, 19, 40, 23, 18, 44, 29, 13, 45, 28, 31, 46, 50, 12, 37, 48, 21, 39, 47, 51, 32, 49, 55, 60, 41, 52, 63, 70, 42, 53, 61, 72, 43, 56, 71, 80, 54, 62, 73, 58, 64, 77, 59, 66, 74, 81, 65, 79
Offset: 0
From _David A. Corneth_, Sep 22 2016: (Start)
Each number can consist of 2^10-1 sets of distinct digits, i.e., classes. For example, 21132 is in the class {1, 2, 3}. We don't include a number without digits. For this sequence, we can also exclude numbers with only the digit 0. This leaves 1022 classes. We create a list with a place for each class containing the least number from that class not already in the sequence.
To illustrate the algorithm used to create the current b-file, we'll (for brevity) assume we've already calculated all terms for n = 1 to 100 and that we already know which classes will be used to compute the next 10 terms, for n = 101 to 110.
These classes are: {0, 1}, {2, 3}, {5, 9}, {7, 9}, {8, 9}, {0, 1, 6}, {0, 1, 7}, {2, 2, 2} and {2, 2, 4} having the values 110, 223, 95, 97, 89, 106, 107, 222 and 224. a(99) = 104 and a(100) = 88, so from those values we may only choose from {223, 95, 97 and 222}. The least value in the list is 95. Therefore, a(101) = 95. The number for the class is now replaced with the next larger number having digits {5, 9} (=A276769(95)), being 559.
(One may see that in the example I only listed 9 classes. Class {8, 9} occurs twice in the example; a(104) = 89 and a(107) = 98.)
From a list of computed values up to some n, the values for classes may be updated to compute further. E.g., to compute a(20000), one may use the b-file to find the least number not already in the sequence for each class and then proceed from a(19998) and a(19999), etc. (End)
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N:= 10^3: # to get all terms before the first > N
for R in combinat:-powerset({$0..9}) minus {{},{$0..9}} do
Lastused[R]:= [];
MR[R]:= Array[0..9];
for i from 1 to nops(R) do MR[R][R[i]]:= i od:
od:
A[0]:= 0: A[1]:= 1:
S:= {0,1}:
for n from 2 to N do
R:= {$0..9} minus (convert(convert(A[n-1],base,10),set) union convert(convert(A[n-2],base,10),set));
L:= Lastused[R];
x:= 0;
while member(x,S) do
for d from 1 do
if d > nops(L) then
if R[1] = 0 then L:= [op(L),R[2]] else L:= [op(L),R[1]] fi;
break
elif L[d] < R[-1] then
L[d]:= R[MR[R][L[d]]+1]; break
else
L[d]:= R[1];
fi
od;
x:= add(L[j]*10^(j-1),j=1..nops(L));
od;
A[n]:= x;
S:= S union {x};
Lastused[R] := L;
od:
seq(A[i],i=0..N); # Robert Israel, Sep 20 2016
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s={0,1};Do[a=s[[-2]];b=s[[-1]];n=2;idab=Union[IntegerDigits[a],IntegerDigits[b]]; While[MemberQ[s,n]|| Intersection[idab,IntegerDigits[n]]!={},n++];AppendTo[s, n],{100}];s
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from itertools import count, islice, product as P
def only(s, D=1): # numbers with >= D digits only from s
yield from (int("".join(p)) for d in count(D) for p in P(s, repeat=d))
def agen(): # generator of terms
aset, an1, an, minan = {0, 1}, 0, 1, 2
yield from [0, 1]
while True:
an1, an, s = an, minan, set(str(an) + str(an1))
use = "".join(c for c in "0123456789" if c not in s)
for an in only(use, D=len(str(minan))):
if an not in aset: break
aset.add(an)
yield an
while minan in aset: minan += 1
print(list(islice(agen(), 75))) # Michael S. Branicky, Jun 30 2022
A239664
a(n) = 1, a(n+1) = smallest number not occurring earlier, having with a(n) neither a common digit nor a common divisor.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 11, 20, 13, 22, 15, 26, 17, 24, 19, 25, 14, 27, 16, 29, 18, 35, 12, 37, 21, 34, 55, 28, 31, 40, 33, 41, 30, 47, 32, 45, 38, 49, 36, 59, 42, 53, 44, 39, 46, 51, 43, 50, 61, 48, 65, 71, 52, 63, 58, 67, 54, 73, 56, 79, 60, 77
Offset: 1
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import Data.List (delete, intersect); import Data.Function (on)
a239664 n = a239664_list !! (n-1)
a239664_list = 1 : f 1 [2..] where
f v ws = g ws where
g (x:xs) = if gcd v x == 1 && ((intersect `on` show) v x == "")
then x : f x (delete x ws) else g xs
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{u=[]; a=1; for(n=1,99, print1(a","); u=setunion(u,[a]); while(#u>1&&u[2]==u[1]+1,u=u[^1]); for(k=u[1]+1,9e9, setsearch(u,k)&&next;gcd(k,a)>1&&next; #setintersect(Set(digits(a)),Set(digits(k)))&&next; a=k; next(2)));a} \\ M. F. Hasler, Sep 17 2016
A276766
a(n) = smallest nonnegative integer not yet in the sequence with no repeated digits and no digits in common with a(n-1), starting with a(0)=0.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23, 14, 20, 13, 24, 15, 26, 17, 25, 16, 27, 18, 29, 30, 12, 34, 19, 28, 31, 40, 21, 35, 41, 32, 45, 36, 42, 37, 46, 38, 47, 39, 48, 50, 43, 51, 49, 52, 60, 53, 61, 54, 62, 57, 63, 58, 64, 59, 67, 80, 56, 70, 65, 71, 68, 72, 69, 73, 81, 74, 82, 75
Offset: 0
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{u=[]; (t(k)=if(#Set(k=digits(k))==#k,k)); a=1; for(n=1, 99, print1(a","); u=setunion(u, [a]); t(u[1])||u[1]++; while(#u>1&&u[2]<=u[1]+1, u=u[^1]); for(k=u[1]+1, 9e9, setsearch(u, k)&&next; (d=t(k))&& !#setintersect(Set(digits(a)), Set(d))&&(a=k)&&next(2))); a} \\ M. F. Hasler, Sep 17 2016
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def ok(s, t): return len(set(t)) == len(t) and len(set(s+t)) == len(s+t)
def agen(): # generator of complete sequence of terms
aset, k, mink, MAX = {0}, 0, 1, 987654321
while True:
if k < MAX: yield k
else: return
k, s = mink, str(k)
MAX = 10**(10-len(s))
while k < MAX and (k in aset or not ok(s, str(k))):
k += 1
aset.add(k)
while mink in aset: mink += 1
print(list(agen())[:73]) # Michael S. Branicky, Jun 30 2022
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