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%I #24 Apr 03 2023 10:36:13
%S 0,0,0,8,22,30,31,35,38,43,48,51
%N Largest possible number of digits in a base n right-truncatable semiprime.
%C Right-truncatable semiprimes are numbers, where the number itself and all numbers obtained by successively removing the rightmost digit are semiprimes. S. S. Gupta found the largest possible right-truncatable base 10 semiprime to be 95861957783594714393831931415189937897 (38 decimal digits). Digit counts for largest possible right-truncatable semiprimes in other bases, found by Hermann Jurksch, are given in this sequence.
%H Shyam Sunder Gupta, <a href="https://t5k.org/curios/page.php?curio_id=6861">The largest right-truncatable semiprime</a>, Prime Curios.
%e There are no right-truncatable semiprimes in bases 2,3 and 4 thus a(2)=a(3)=a(4)=0;
%e The examples give the smallest base n semiprimes of maximum digit count, found by H. Jurksch:
%e a(5)=8: 42143413
%e a(6)=22: 4223145115415551545111
%e a(7)=30: 644324264233631242462662622646
%e a(8)=31: 4267773725372537135533515117773
%e a(9)=35: 43741424882428682844851886888222774
%e a(10)=38: 93359393537779942973989331953313839313
%e a(11)=43: 4567476a2738a828994aa851a116aa886a95686a231
%e a(12)=48: 43a2971ba155719171a2b1b97777775b779a732b755572b7
%e a(13)=51: 9114448462c6c46b3c9937446466b43686a246686667324c6a2
%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 d, s = 0, [(i,) for i in range(n) if semiprime(fromdigits((i,), n))]
%o while len(s) > 0:
%o cands = set(t+(d,) for t in s for d in tuple(range(n)))
%o d, s = d+1, [c for c in cands if semiprime(fromdigits(c, n))]
%o return d
%o print([a(n) for n in range(2, 8)]) # _Michael S. Branicky_, Aug 04 2022
%Y Cf. A001358, A085733, A213018.
%K nonn,base,hard,more
%O 2,4
%A _Hugo Pfoertner_, Jun 07 2012
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