A323137 Largest prime that is both left-truncatable and right-truncatable in base n.

23, 11, 67, 839, 37, 1867, 173, 739397, 79, 105691, 379, 37573, 647, 3389, 631, 202715129, 211, 155863, 1283, 787817, 439, 109893629, 577, 4195880189, 1811, 14474071, 379, 21335388527, 2203, 1043557, 2939, 42741029, 2767, 50764713107, 853, 65467229, 4409, 8524002457

Offset: 3

Author

Felix Fröhlich, Jan 05 2019

Keywords

• nonn
• base

Examples

			For n = 12: 105691 is 511B7 in base 12. Successively removing the leftmost digit yields the base-12 numbers 11B7, 1B7, B7 and 7. When converted to base 10, these are 2011, 283, 139 and 7, respectively, all primes. Successively removing the rightmost digit yields the base-12 numbers 511B, 511, 51 and 5. When converted to base 10, these are 8807, 733, 61 and 5, respectively, all primes. Since no larger prime with this property in base 12 exists (as proven by Daniel Suteu), a(12) = 105691.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 3..64
• Wikipedia, Truncatable prime

Crossrefs

Cf. A020994, A023107, A076586, A076623, A103443, A323390, A323396.

Programs

PARI

digitsToNum(d, base) = sum(k=1, #d, base^(k-1) * d[k]);
isLeftTruncatable(d, base) = my(ok=1); for(k=1, #d, if(!isprime(digitsToNum(d[1..k], base)), ok=0; break)); ok;
generateFromPrefix(p, base) = my(seq = [p]); for(n=1, base-1, my(t=concat(n, p)); if(isprime(digitsToNum(t, base)), seq=concat(seq, select(v -> isLeftTruncatable(v, base), generateFromPrefix(t, base))))); seq;
bothTruncatablePrimesInBase(base) = my(t=[]); my(P=primes(primepi(base-1))); for(k=1, #P, t=concat(t, generateFromPrefix([P[k]], base))); vector(#t, k, digitsToNum(t[k], base));
a(n) = vecmax(bothTruncatablePrimesInBase(n)); \\ for n>=3; Daniel Suteu, Jan 22 2019

Formula

a(n) <= min(A023107(n), A103443(n)). - Daniel Suteu, Feb 24 2019

Extensions

a(17)-a(40) from Daniel Suteu, Jan 11 2019