A107353 -id:A107353 - cp's OEIS frontend

A107801 a(1) = prime(1), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

2, 23, 3, 13, 11, 17, 7, 37, 31, 19, 29, 59, 5, 53, 43, 41, 47, 67, 61, 71, 73, 79, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov & Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. If it exists, N > 10^1000. - Charles R Greathouse IV, Jul 19 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A107353.

Other cases of seed: A107802 (a(1) = 3), A107803 (a(1) = 5), A107804 (a(1) = 7), A107805 (a(1) = 11), A107806 (a(1) = 13), A107807 (a(1) = 17), A107808 (a(1) = 19), A107809 (a(1) = 23), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41), A107814 (a(1) = 43).

import Data.List (intersect, delete)
a107801 n = a107801_list !! (n-1)
a107801_list = 2 : f 2 (tail a000040_list) where
   f x ps = g ps where
     g (q:qs) | null (show x `intersect` show q) = g qs
              | otherwise                        = q : f q (delete q ps)
-- Reinhard Zumkeller, Mar 31 2012

p=Prime[1];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

common(a,b)=a=vecsort(eval(Vec(Str(a))),,8);b=vecsort(eval(Vec(Str(b))),,8);#a+#b>#vecsort(concat(a,b),,8)
in(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lista(nn) = {my(v=[2]); for(n=2, nn, forprime(p=2, default(primelimit), if(!in(v,p)&&common(v[#v],p), v=concat(v,p); break))); v; }
\\ Charles R Greathouse IV, Jul 20 2011

a(n) ~ n log n. - Charles R Greathouse IV, Jul 19 2011

For n>=29, A(107800+i)(n) = A(107800+j)(n), 1 <= i < j <= 14. - Vladimir Shevelev, Mar 18 2012

A107814 a(1) = prime(14), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

Original entry on oeis.org

43, 3, 13, 11, 17, 7, 37, 23, 2, 29, 19, 31, 41, 47, 67, 61, 71, 73, 53, 5, 59, 79, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov and Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. - Charles R Greathouse IV, Jul 20 2011

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A107353.

Other cases of seed: A107801 (a(1) = 2), A107802 (a(1) = 3), A107803 (a(1) = 5), A107804 (a(1) = 7), A107805 (a(1) = 11), A107806 (a(1) = 13), A107807 (a(1) = 17), A107808 (a(1) = 19), A107809 (a(1) = 23), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41).

Cands:= subsop(14=NULL, [seq(ithprime(i),i=1..1000)]):
S:= map(t -> convert(convert(t,base,10),set), Cands):
R:= 43: x:= 43: xs:= {3,4}:
for n from 2 to 100 do
  found:= false;
  for i from 1 do
    if S[i] intersect xs <> {} then
      R:= R, Cands[i];
      x:= Cands[i];
      xs:= S[i];
      Cands:= subsop(i=NULL,Cands);
      S:= subsop(i=NULL,S);
      found:= true;
      break
    fi
  od;
  if not found then break fi;
od:
R; # Robert Israel, Dec 16 2024

p=Prime[14];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

a(n) ~ n log n. - Charles R Greathouse IV, Jul 20 2011

A184992 a(n) is the least positive integer not occurring earlier that shares a digit with a(n-1); a(1)=1.

Original entry on oeis.org

1, 10, 11, 12, 2, 20, 21, 13, 3, 23, 22, 24, 4, 14, 15, 5, 25, 26, 6, 16, 17, 7, 27, 28, 8, 18, 19, 9, 29, 32, 30, 31, 33, 34, 35, 36, 37, 38, 39, 43, 40, 41, 42, 44, 45, 46, 47, 48, 49, 54, 50, 51, 52, 53, 55, 56, 57, 58, 59, 65, 60, 61, 62, 63, 64, 66, 67, 68, 69, 76, 70, 71, 72, 73

Offset: 1

Eric Angelini, Dec 22 2011

A permutation of the positive integers.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A162501, A076654, A130571.
a(n) = A107353(n) for n>=3. - Alois P. Heinz, Dec 22 2011
Cf. A227118 (inverse); A067581.

import Data.List (delete, intersect); import Data.Function (on)
a184992 n = a184992_list !! (n-1)
a184992_list = 1 : f 1 [2..] where
   f u vs = v : f v (delete v vs)
     where v : _ = filter (not . null . (intersect `on` show) u) vs
-- Reinhard Zumkeller, Jul 01 2013

FromDigits /@ Nest[Function[a, Append[a, Block[{k = 2, d}, While[Nand[FreeQ[a, #], IntersectingQ[a[[-1]], #]] &@ Set[d, IntegerDigits@ k], k++]; d]]], {{1}}, 73] (* Michael De Vlieger, Mar 17 2018 *)

A184992(n,show=0)={my(a=1,u=2^1);for(k=2,n,show && print1(a",");a=Set(Vec(Str(a))); for(j=2,9e9,bittest(u,j) && next;setintersect(Set(Vec(Str(j))),a) || next; u+=2^a=j; break));a}  \\ M. F. Hasler, Dec 22 2011

from itertools import count, islice
def agen(): # generator of terms
    an, aset, mink = 1, {1}, 1
    while True:
        yield an
        digset = set(str(an))
        an = next(k for k in count(mink) if k not in aset and set(str(k))&digset)
        aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 74))) # Michael S. Branicky, Oct 03 2024

A107809 a(1) = prime(9), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

Original entry on oeis.org

23, 2, 29, 19, 11, 13, 3, 31, 17, 7, 37, 43, 41, 47, 67, 61, 71, 73, 53, 5, 59, 79, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov & Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. - Charles R Greathouse IV, Jul 20 2011

Cf. A107353.

Other cases of seed: A107801 (a(1) = 2), A107802 (a(1) = 3), A107803 (a(1) = 5), A107804 (a(1) = 7), A107805 (a(1) = 11), A107806 (a(1) = 13), A107807 (a(1) = 17), A107808 (a(1) = 19), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41), A107814 (a(1) = 43).

p=Prime[9];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

common(a,b)=a=vecsort(eval(Vec(Str(a))),,8);b=vecsort(eval(Vec(Str(b))),,8);#a+#b>#vecsort(concat(a,b),,8)
in(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lista(nn) = {my(v=[23]); for(n=2, nn, forprime(p=2, default(primelimit), if(!in(v,p)&&common(v[#v],p), v=concat(v,p); break))); v; }
\\ Charles R Greathouse IV, Jul 20 2011

a(n) ~ n log n. - Charles R Greathouse IV, Jul 20 2011

A107802 a(1) = prime(2), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

Original entry on oeis.org

3, 13, 11, 17, 7, 37, 23, 2, 29, 19, 31, 41, 43, 47, 67, 61, 71, 73, 53, 5, 59, 79, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov & Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. - Charles R Greathouse IV, Jul 20 2011

Cf. A107353.

Other cases of seed: A107801 (a(1) = 2), A107803 (a(1) = 5), A107804 (a(1) = 7), A107805 (a(1) = 11), A107806 (a(1) = 13), A107807 (a(1) = 17), A107808 (a(1) = 19), A107809 (a(1) = 23), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41), A107814 (a(1) = 43).

p=Prime[2];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

common(a,b)=a=vecsort(eval(Vec(Str(a))),,8);b=vecsort(eval(Vec(Str(b))),,8);#a+#b>#vecsort(concat(a,b),,8)
in(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lista(nn) = {my(v=[3]); for(n=2, nn, forprime(p=2, default(primelimit), if(!in(v,p)&&common(v[#v],p), v=concat(v,p); break))); v; }
\\ Charles R Greathouse IV, Jul 20 2011

a(n) ~ n log n. - Charles R Greathouse IV, Jul 20 2011

A107803 a(1) = prime(3), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

Original entry on oeis.org

5, 53, 3, 13, 11, 17, 7, 37, 23, 2, 29, 19, 31, 41, 43, 47, 67, 61, 71, 73, 79, 59, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov & Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. - Charles R Greathouse IV, Jul 20 2011

Cf. A107353.

Other cases of seed: A107801 (a(1) = 2), A107802 (a(1) = 3), A107804 (a(1) = 7), A107805 (a(1) = 11), A107806 (a(1) = 13), A107807 (a(1) = 17), A107808 (a(1) = 19), A107809 (a(1) = 23), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41), A107814 (a(1) = 43).

p=Prime[3];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

common(a,b)=a=vecsort(eval(Vec(Str(a))),,8);b=vecsort(eval(Vec(Str(b))),,8);#a+#b>#vecsort(concat(a,b),,8)
in(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lista(nn) = {my(v=[5]); for(n=2, nn, forprime(p=2, default(primelimit), if(!in(v,p)&&common(v[#v],p), v=concat(v,p); break))); v; }
\\ Charles R Greathouse IV, Jul 20 2011

a(n) ~ n log n. - Charles R Greathouse IV, Jul 20 2011

A107804 a(1) = prime(4), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

Original entry on oeis.org

7, 17, 11, 13, 3, 23, 2, 29, 19, 31, 37, 43, 41, 47, 67, 61, 71, 73, 53, 5, 59, 79, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov & Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. - Charles R Greathouse IV, Jul 20 2011

Cf. A107353.

Other cases of seed: A107801 (a(1) = 2), A107802 (a(1) = 3), A107803 (a(1) = 5), A107805 (a(1) = 11), A107806 (a(1) = 13), A107807 (a(1) = 17), A107808 (a(1) = 19), A107809 (a(1) = 23), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41), A107814 (a(1) = 43).

p=Prime[4];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

common(a,b)=a=vecsort(eval(Vec(Str(a))),,8);b=vecsort(eval(Vec(Str(b))),,8);#a+#b>#vecsort(concat(a,b),,8)
in(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lista(nn) = {my(v=[7]); for(n=2, nn, forprime(p=2, default(primelimit), if(!in(v,p)&&common(v[#v],p), v=concat(v,p); break))); v; }
\\ Charles R Greathouse IV, Jul 20 2011

a(n) ~ n log n. - Charles R Greathouse IV, Jul 20 2011

A107805 a(1) = prime(5), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

Original entry on oeis.org

11, 13, 3, 23, 2, 29, 19, 17, 7, 37, 31, 41, 43, 47, 67, 61, 71, 73, 53, 5, 59, 79, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov & Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. - Charles R Greathouse IV, Jul 20 2011

Cf. A107353.

Other cases of seed: A107801 (a(1) = 2), A107802 (a(1) = 3), A107803 (a(1) = 5), A107804 (a(1) = 7), A107806 (a(1) = 13), A107807 (a(1) = 17), A107808 (a(1) = 19), A107809 (a(1) = 23), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41), A107814 (a(1) = 43).

p=Prime[5];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

common(a,b)=a=vecsort(eval(Vec(Str(a))),,8);b=vecsort(eval(Vec(Str(b))),,8);#a+#b>#vecsort(concat(a,b),,8)
in(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lista(nn) = {my(v=[11]); for(n=2, nn, forprime(p=2, default(primelimit), if(!in(v, p)&&common(v[#v], p), v=concat(v, p); break))); v; }
\\ Charles R Greathouse IV, Jul 20 2011

a(n) ~ n log n. - Charles R Greathouse IV, Jul 20 2011

A107806 a(1) = prime(6), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

Original entry on oeis.org

13, 3, 23, 2, 29, 19, 11, 17, 7, 37, 31, 41, 43, 47, 67, 61, 71, 73, 53, 5, 59, 79, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov & Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. - Charles R Greathouse IV, Jul 20 2011

Cf. A107353.

Other cases of seed: A107801 (a(1) = 2), A107802 (a(1) = 3), A107803 (a(1) = 5), A107804 (a(1) = 7), A107805 (a(1) = 11), A107807 (a(1) = 17), A107808 (a(1) = 19), A107809 (a(1) = 23), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41), A107814 (a(1) = 43).

p=Prime[6];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

common(a,b)=a=vecsort(eval(Vec(Str(a))),,8);b=vecsort(eval(Vec(Str(b))),,8);#a+#b>#vecsort(concat(a,b),,8)
in(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lista(nn) = {my(v=[13]); for(n=2, nn, forprime(p=2, default(primelimit), if(!in(v,p)&&common(v[#v],p), v=concat(v,p); break))); v; }
\\ Charles R Greathouse IV, Jul 20 2011

a(n) ~ n log n. - Charles R Greathouse IV, Jul 20 2011

A107807 a(1) = prime(7), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

Original entry on oeis.org

17, 7, 37, 3, 13, 11, 19, 29, 2, 23, 31, 41, 43, 47, 67, 61, 71, 73, 53, 5, 59, 79, 89, 83, 103, 101, 107, 97, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Zak Seidov & Eric Angelini, May 24 2005

a(n) = prime(n) for almost all n. Probably a(n) = prime(n) for all n > N for some N, but N must be very large. - Charles R Greathouse IV, Jul 20 2011

Cf. A107353.

Other cases of seed: A107801 (a(1) = 2), A107802 (a(1) = 3), A107803 (a(1) = 5), A107804 (a(1) = 7), A107805 (a(1) = 11), A107806 (a(1) = 13), A107808 (a(1) = 19), A107809 (a(1) = 23), A107810 (a(1) = 29), A107811 (a(1) = 31), A107812 (a(1) = 37), A107813 (a(1) = 41), A107814 (a(1) = 43).

p=Prime[7];b={p};d=p;Do[Do[r=Prime[c];If[FreeQ[b, r]&&Intersection@@IntegerDigits/@{d, r}=!={}, b=Append[b, r];d=r;Break[]], {c, 1000}], {k, 60}];b

common(a,b)=a=vecsort(eval(Vec(Str(a))),,8);b=vecsort(eval(Vec(Str(b))),,8);#a+#b>#vecsort(concat(a,b),,8)
in(v,x)=for(i=1,#v,if(v[i]==x,return(1)));0
lista(nn) = {my(v=[17]); for(n=2, nn, forprime(p=2, default(primelimit), if(!in(v,p)&&common(v[#v],p), v=concat(v,p); break))); v; }
\\ Charles R Greathouse IV, Jul 20 2011

a(n) ~ n log n. - Charles R Greathouse IV, Jul 20 2011

cp's OEIS Frontend

A107801 a(1) = prime(1), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

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A107814 a(1) = prime(14), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

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A184992 a(n) is the least positive integer not occurring earlier that shares a digit with a(n-1); a(1)=1.

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A107809 a(1) = prime(9), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

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A107802 a(1) = prime(2), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

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A107803 a(1) = prime(3), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

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A107804 a(1) = prime(4), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

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A107805 a(1) = prime(5), for n >= 2, a(n) is the smallest prime not previously used which contains a digit from a(n-1).

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