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%I #58 Jan 02 2017 01:55:35
%S 11,13,31,17,71,19,97,73,37,79,907,7,701,101,103,3,307,709,911,113,
%T 131,311,1103,1031,313,137,373,733,331,317,173,739,397,971,719,191,
%U 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
%N 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.
%C 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
%C 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
%C 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
%H M. F. Hasler, <a href="/A089755/b089755.txt">Table of n, a(n) for n = 1..250</a>
%e After a(1) = 11, the next term must start with 1. 11 is already present, so 13 is next.
%e After 13, the next term starts with 3, but it cannot be just 3 as that is a single digit.
%e 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.
%e 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 occured, it's 307.
%o (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
%Y Cf. A089756.
%Y See A262282 and A262283 for other versions.
%K base,easy,nonn
%O 1,1
%A _Amarnath Murthy_, Nov 22 2003
%E Edited by _Franklin T. Adams-Watters_, Sep 18 2015
%E Definition reworded and data corrected and extended by _M. F. Hasler_, Sep 18 2015
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