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a(n) is the smallest odd prime p such that 2^n*p has n digits but has at most two distinct digits; or 0 if no such prime exists.
1

%I #5 Apr 03 2023 10:36:11

%S 3,3,29,101,691,15467,39023,71023,437977,4344227,21158903,109739989,

%T 344590189,2956838897,6781690193,0,85533990571,3390460543777,0,

%U 53936545044581,0,0,5298071316879193,0,168548719780643483

%N a(n) is the smallest odd prime p such that 2^n*p has n digits but has at most two distinct digits; or 0 if no such prime exists.

%C Andrew Rupinski showed that a(95) exists (see the links below).

%H Prime Puzzles & Problems Connection, <a href="http://www.primepuzzles.net/puzzles/puzz_376.htm">Puzzle 376. n=p*2^x</a>.

%H Andrew Rupinski, <a href="https://t5k.org/curios/page.php?number_id=2688&amp;submitter=Rupinski">Prime Curios!</a>.

%e a(14)=2956838897 because 2^14*2956838897=48444848488448 has 14 digits with two distinct digits and 2956838897 is the smallest prime p such that 2^14*p has these properties.

%t a[1]=3; a[n_]:=(For[m=Floor[5^(n-1)/2],!(PrimeQ[m]&&Length[Union[ IntegerDigits[2^n*m]]]==2&&Length[IntegerDigits[2^n*m]]==n),m++ ]; m);Do[Print[a[n]],{n,14}]

%Y Cf. A124245.

%K nonn,base

%O 1,1

%A _Farideh Firoozbakht_, Oct 25 2006

%E Edited by _Don Reble_, Oct 29 2006