A089755 a(1)=11; for n>1, a(n) is the smallest prime not occurring earlier beginning with a(n-1) without its first digit. Single-digit primes are not allowed unless they arise from the previous term as multi-digit number with leading zero(s) (i.e., a(n-1) has 0 as second digit) which are remembered for the subsequent left-truncations.

11, 13, 31, 17, 71, 19, 97, 73, 37, 79, 907, 7, 701, 101, 103, 3, 307, 709, 911, 113, 131, 311, 1103, 1031, 313, 137, 373, 733, 331, 317, 173, 739, 397, 971, 719, 191, 919, 193, 937, 379, 797, 977, 773, 7307, 3079, 7901, 9011, 1109, 109, 929, 29, 941, 41, 107, 727, 271, 7103, 1033, 337, 3701, 7013

Offset: 1

Author

Amarnath Murthy, Nov 22 2003

Keywords

• base
• easy
• nonn

Comments

Single-digit numbers are not permitted. However, the sequence is calculated allowing leading zeros. See examples. Uniqueness is still determined by the numerical value: 1013, if it appears, cannot be followed by 013 because 13 is already present. - Franklin T. Adams-Watters, Sep 18 2015

The first digit 2 appears in a(50) = 929 following 109 (since 907, 911, 919 occur earlier), and the first digit 4 appears after a(51) = 29 in a(52) = 941 (because a(39) = 937). The first digit 5 appears in a(199), a 55 digit prime which ends in ...51. The digit 8 appears first in a(230), a 95 digit prime ending in ...81. - M. F. Hasler, Sep 18 2015

As noted by Alois P. Heinz in an email, it appears certain that most primes do not occur in this sequence: "Most primes will never show up, because the net length of terms increases with accelerated rate. There are good reasons for that, but I don't have time to write this up now. But see the numerical evidence." In particular, since the density of primes is zero, there comes a point where it is more likely that two (or more) digits need to be added than that a digit can be removed. From that point, the sequence grows at an ever-increasing rate. - Franklin T. Adams-Watters, Sep 19 2015

Examples

			After a(1) = 11, the next term must start with 1. 11 is already present, so 13 is next.
After 13, the next term starts with 3, but it cannot be just 3 as that is a single digit.
After 103, dropping the lead digit leaves 03, and 3 has not occurred before, so that is the next term. As sequences in the OEIS are sequences of numbers, not numerals, this appears in the sequence as 3.
The subsequent term will be the next prime having as first digits the preceding 03 with the initial 0 removed (but not the 3 removed). Thus, it must start with 3, and since 3, 31 and 37 have already occurred, it's 307.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A089756.

See A262282 and A262283 for other versions.

Programs

PARI

A089755(N,D=-1,u=[],a=11)={my(d); for(n=1,N, print1(a","); D>=0&&setsearch(Set(digits(a)),D)&&return([n,a]);u=setunion(u,[a]);a>9&&isprime(a%=10^d=#Str(a)-1)&&d>1&&!setsearch(u,a)&&next;for(d=1,999,a*=10;forstep(i=1,10^d-1,2,ispseudoprime(a+i)&&a+i>9&&!setsearch(u,a+i)&&(a+=i)&&next(3))));a} \\ If a 2nd argument D is given, then returns [n, a(n)] for the first a(n) having the digit D, if this occurs before the limit N. M. F. Hasler, Sep 18 2015

Extensions

Edited by Franklin T. Adams-Watters, Sep 18 2015

Definition reworded and data corrected and extended by M. F. Hasler, Sep 18 2015