A213018 Largest possible right-truncatable base n semiprime, written in decimal notation.

349859, 96614184696363331, 21453921664462866568480385, 5396625577204731352098054139, 1230847457959658263441326143300761, 95861957783594714393831931415189937897, 246968512564969427282294385793684699270364003, 2275670244821939317343219562642735197101789412250091, 452359410421075824795509870868069265597540337861667320077

Offset: 5

Author

Hugo Pfoertner, Jun 26 2012

Keywords

• nonn
• base
• hard

Comments

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.

Examples

			a(5)=349859=42143414 in base 5 = 89*3931
4214341 in base 5 = 69971 = 11*6361
421434 in base 5 = 13994 = 2*6997
42143 in base 5 = 2798 = 2*1399
4214 in base 5 = 559 = 13*43
421 in base 5 = 111 = 3*37
42 in base 5 = 22 = 2*11
4 in base 5 = 4 = 2*2
a(6)=4223145115415551545111 in base 6
a(7)=644324264233631242462662622646 in base 7
a(8)=4267773725372537135533515117773 in base 8
a(9)=43741424882428682844851886888222774 in base 9
a(10)=95861957783594714393831931415189937897 in base 10
a(11)=4567476a2738a828994aa851a116aa886a95686a231 in base 11
a(12)=43a2971ba155719171a2b1b97777775b779a732b755572b7 in base 12
a(13)=9114448462c6c46b3c9937446466b43686a24668666732c4356 in base 13
		

Crossrefs

Cf. A001358, A085733, A213017.

Programs

Python

from sympy import factorint
def fromdigits(t, b): return sum(b**i*di for i, di in enumerate(t[::-1]))
def semiprime(n): return sum(factorint(n).values()) == 2
def a(n):
    m, s = 0, [(i,) for i in range(n) if semiprime(fromdigits((i,), n))]
    while len(s) > 0:
        m = fromdigits(max(s), n)
        cands = set(t+(d,) for t in s for d in tuple(range(n)))
        s = [c for c in cands if semiprime(fromdigits(c, n))]
    return m
print([a(n) for n in range(5, 8)]) # Michael S. Branicky, Aug 04 2022