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The smallest number whose prime factor concatenation when written in all bases k, where k = 2...n, contains all digits 0,1,...,(k-1).
3

%I #8 Sep 11 2024 10:07:07

%S 2,6,38,190,5662,39255,1206042,22481939,392228153,6329975006

%N The smallest number whose prime factor concatenation when written in all bases k, where k = 2...n, contains all digits 0,1,...,(k-1).

%C Up to a(11) all terms have prime factors whose concatenation length in base n is n, the minimum possible value. Is this true for all a(n)?

%e a(7) = 39255 as 39255 = 3*5*2617 = 11_2*101_2*101000111001_2 = "11101101000111001" which contains all digits 0...1 = 10_3*12_3*10120221_3 = "101210120221" which contains all digits 0...2, 3_4*11_4*220321_4 = "311220321" which contains all digits 0...3 = 3_5*10_5*40432_5 = "31040432" which contains all digits 0...4 = 3_6*5_6*20041_6 = "3520041" which contains all digits 0...5 = 3_7*5_7*10426_7 = "3510426" which contains all digits 0...6.

%Y Cf. A372309, A373645, A055085, A027748.

%K nonn,base,more

%O 2,1

%A _Scott R. Shannon_, Sep 09 2024