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a(n) is the least positive integer k such that k*(k+1)*...*(k+n-1) does not contain the digit 2, or -1 if there is no such k.
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%I #4 Feb 16 2023 05:40:31

%S 1,2,1,3,7,2,1,3,3,2,1,1,3,2,1,5,10,10,10,17,4,8,38,38,19,17,2,1,1,3

%N a(n) is the least positive integer k such that k*(k+1)*...*(k+n-1) does not contain the digit 2, or -1 if there is no such k.

%C a(n) is the least positive integer k such that (k+n-1)!/(k-1)! does not contain the digit 2, or -1 if there is no such k.

%C a(32) = 13.

%C Conjecture: a(n) = -1 for n = 31 and all n >= 33.

%e a(4) = 3 because 3*4*5*6 = 360 does not contain the digit 2, while 1*2*3*4 = 24 and 2*3*4*5 = 120 do.

%p f:= proc(n) local k,t;

%p t:= n!;

%p for k from 1 to 100000 do

%p if not member(2,convert(t,base,10)) then return k fi;

%p t:= t*(n+k)/k;

%p od:

%p -1

%p end proc:

%p map(f, [$1..32]);

%Y Cf. A173333.

%K nonn,base,more

%O 1,2

%A _Robert Israel_, Feb 12 2023