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A350153 Prime numbers created by concatenating all numbers 1 through k for some k > 1, then continuing to concatenate all numbers from k-1 towards 1. Primes are added to the sequence as they are found as k increases.

%I A350153 #65 May 11 2024 21:55:51
%S A350153 12343,1234543,12345678910987,12345678910987654321,
%T A350153 12345678910111213141516171819202122212019181716151413,
%U A350153 12345678910111213141516171819202122232425262728293029
%N A350153 Prime numbers created by concatenating all numbers 1 through k for some k > 1, then continuing to concatenate all numbers from k-1 towards 1. Primes are added to the sequence as they are found as k increases.
%C A350153 A173426(n) is the concatenation of all numbers from 1 up to k and then back down to 1. The prime terms of A173426 have been called "memorable primes" (see the Numberphile video). These "unmemorable primes" are a superset created by concatenating 1..k in ascending order followed by concatenating the numbers k-1..1 in descending order. Any primes found during either concatenation process are added to the sequence (e.g., k = 5, 1234543 is included. If 12345 were prime, it would be included as well).
%H A350153 Patrick Quam, <a href="/A350153/b350153.txt">Table of n, a(n) for n = 1..29</a>
%H A350153 Brady Haran and N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=vKlVNFOHJ9I">The Most Wanted Prime Number</a>, Numberphile video (2021).
%e A350153 For k=10, the first prime obtained by concatenating the numbers 1..10 and then concatenating the first one or more numbers from 9..1 is 12345678910987.
%p A350153 select(isprime, [seq(seq(parse(cat($1..n, n-i$i=1..t)),
%p A350153                 t=0..n-1), n=1..30)])[];  # _Alois P. Heinz_, Dec 19 2021
%t A350153 lst={};Table[s=Flatten[IntegerDigits/@Range@n];k=n-1;
%t A350153 While[k!=-1,If[PrimeQ[p=FromDigits@s],AppendTo[lst,p]];s=Join[s,IntegerDigits@k];k--],{n,100}];lst (* _Giorgos Kalogeropoulos_, Dec 17 2021 *)
%o A350153 (Python)
%o A350153 from itertools import count, chain, islice, accumulate
%o A350153 from sympy import isprime
%o A350153 def A350153gen(): return filter(lambda p:isprime(p),(int(s) for n in count(1) for s in accumulate(str(d) for d in chain(range(1,n+1),range(n-1,0,-1)))))
%o A350153 A350153_list = list(islice(A350153gen(),20)) # _Chai Wah Wu_, Dec 20 2021
%Y A350153 Cf. A048847, A007908, A173426, A260343, A323532.
%K A350153 nonn,base
%O A350153 1,1
%A A350153 _Patrick Quam_, Dec 16 2021