%I #21 Mar 18 2019 08:09:05
%S 1,4,3,3,2,2,3,2,2,1,2,1,2,2,2,1,2,1,6,6,1,5,3,4,2,3,1,5,1,6,2,2,5,1,
%T 2,4,4,1,3,4,3,4,1,3,2,3,2,2,1,5,2,2,2,1,4,1,4,3,3,3,1,3,3,3,1,2,1,2,
%U 4,4,2,1,2,2,4,1,3,3,3,4,1,3,3,2,3,2,2
%N Start with n; add to it any of its digits; repeat; a(n) = minimal number of steps needed to reach a prime greater than n.
%C Is it a theorem that a(n) aways exists?
%C Yes: as long as nonzero digits are used, eventually you reach a number x starting with 10, large enough that there is a prime between x and 3*x/2. All the numbers from x to 3*x/2 start with 1, so if you use the digit 1 you will eventually reach a prime. - _Robert Israel_, Mar 17 2019
%C A variant of this (A241181) sets a(n) = 0 if n is already a prime.
%D Eric Angelini, Posting to Sequence Fans Mailing List, Apr 20 2014
%H Hiroaki Yamanouchi, <a href="/A241180/b241180.txt">Table of n, a(n) for n = 1..100000</a>
%e Examples, in condensed notation:
%e 1+1=2
%e 2+2=4+4=8+8=16+1=17
%e 3+3=6+6=12+1=13
%e 4+4=8+8=16+1=17
%e 5+5=10+1=11
%e 6+6=12+1=13
%e 7+7=14+4=18+1=19
%e 8+8=16+1=17
%e 9+9=18+1=19
%e 10+1=11
%e 11+1=12+1=13
%e 12+1=13
%e 13+3=16+1=17
%e 14+4=18+1=19
%e 15+1=16+1=17
%e 16+1=17
%e 17+1=18+1=19
%e 18+1=19
%e 19+9=28+8=36+3=39+9=48+8=56+5=61
%e 20+2=22+2=24+2=26+6=32+2=34+3=37
%e ...
%p g:= proc(n,Nmax) option remember; local L,d,t;
%p if isprime(n) then return 0 fi;
%p if n > Nmax then return infinity fi;
%p L:= convert(convert(n,base,10),set) minus {0};
%p 1 + min(seq(procname(n+d),d=L));
%p end proc:
%p f:= proc(n,Nmax) local L,d,t;
%p L:= convert(convert(n,base,10),set) minus {0};
%p 1 + min(seq(g(n+d, Nmax),d=L))
%p end proc:
%p map(f, [$1..200], 1000); # _Robert Israel_, Mar 17 2019
%t A241180[n_] := Module[{c, nx},
%t c = 1; nx = n;
%t While[ !
%t AnyTrue[nx = Flatten[nx + IntegerDigits[nx]],
%t PrimeQ [#] && # > n &], c++];
%t Return[c]];
%t Table[A241180[i], {i, 100}] (* _Robert Price_, Mar 17 2019 *)
%Y Related sequences: A241173, A241174, A241175, A241176, A241177, A241178, A241179, A241180, A241181, A241182, A241183.
%K easy,nonn,base
%O 1,2
%A _N. J. A. Sloane_, Apr 23 2014
%E a(23)-a(87) from _Hiroaki Yamanouchi_, Sep 05 2014
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