A213017 Largest possible number of digits in a base n right-truncatable semiprime.

0, 0, 0, 8, 22, 30, 31, 35, 38, 43, 48, 51

Offset: 2

Author

Hugo Pfoertner, Jun 07 2012

Keywords

• nonn
• base
• hard
• more

Comments

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.

Examples

			There are no right-truncatable semiprimes in bases 2,3 and 4 thus a(2)=a(3)=a(4)=0;
The examples give the smallest base n semiprimes of maximum digit count, found by H. Jurksch:
a(5)=8: 42143413
a(6)=22: 4223145115415551545111
a(7)=30: 644324264233631242462662622646
a(8)=31: 4267773725372537135533515117773
a(9)=35: 43741424882428682844851886888222774
a(10)=38: 93359393537779942973989331953313839313
a(11)=43: 4567476a2738a828994aa851a116aa886a95686a231
a(12)=48: 43a2971ba155719171a2b1b97777775b779a732b755572b7
a(13)=51: 9114448462c6c46b3c9937446466b43686a246686667324c6a2
		

Links

• Shyam Sunder Gupta, The largest right-truncatable semiprime, Prime Curios.

Crossrefs

Cf. A001358, A085733, A213018.

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):
    d, s = 0, [(i,) for i in range(n) if semiprime(fromdigits((i,), n))]
    while len(s) > 0:
        cands = set(t+(d,) for t in s for d in tuple(range(n)))
        d, s = d+1, [c for c in cands if semiprime(fromdigits(c, n))]
    return d
print([a(n) for n in range(2, 8)]) # Michael S. Branicky, Aug 04 2022