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Largest possible right-truncatable base n semiprime, written in decimal notation.
2

%I #18 Aug 05 2022 07:46:03

%S 349859,96614184696363331,21453921664462866568480385,

%T 5396625577204731352098054139,1230847457959658263441326143300761,

%U 95861957783594714393831931415189937897,246968512564969427282294385793684699270364003,2275670244821939317343219562642735197101789412250091,452359410421075824795509870868069265597540337861667320077

%N Largest possible right-truncatable base n semiprime, written in decimal notation.

%C For the definition of a right-truncatable semiprime, see A213017. The process of truncating at the right end of the digit string has to be applied to the base-n representation given in the examples. a(10) was found by S.S. Gupta. All other terms have been computed by Hermann Jurksch.

%e a(5)=349859=42143414 in base 5 = 89*3931

%e 4214341 in base 5 = 69971 = 11*6361

%e 421434 in base 5 = 13994 = 2*6997

%e 42143 in base 5 = 2798 = 2*1399

%e 4214 in base 5 = 559 = 13*43

%e 421 in base 5 = 111 = 3*37

%e 42 in base 5 = 22 = 2*11

%e 4 in base 5 = 4 = 2*2

%e a(6)=4223145115415551545111 in base 6

%e a(7)=644324264233631242462662622646 in base 7

%e a(8)=4267773725372537135533515117773 in base 8

%e a(9)=43741424882428682844851886888222774 in base 9

%e a(10)=95861957783594714393831931415189937897 in base 10

%e a(11)=4567476a2738a828994aa851a116aa886a95686a231 in base 11

%e a(12)=43a2971ba155719171a2b1b97777775b779a732b755572b7 in base 12

%e a(13)=9114448462c6c46b3c9937446466b43686a24668666732c4356 in base 13

%o (Python)

%o from sympy import factorint

%o def fromdigits(t, b): return sum(b**i*di for i, di in enumerate(t[::-1]))

%o def semiprime(n): return sum(factorint(n).values()) == 2

%o def a(n):

%o m, s = 0, [(i,) for i in range(n) if semiprime(fromdigits((i,), n))]

%o while len(s) > 0:

%o m = fromdigits(max(s), n)

%o cands = set(t+(d,) for t in s for d in tuple(range(n)))

%o s = [c for c in cands if semiprime(fromdigits(c, n))]

%o return m

%o print([a(n) for n in range(5, 8)]) # _Michael S. Branicky_, Aug 04 2022

%Y Cf. A001358, A085733, A213017.

%K nonn,base,hard

%O 5,1

%A _Hugo Pfoertner_, Jun 26 2012