%I #25 Mar 23 2024 20:35:31
%S 34,258,2147,11582,62192,274810,1500309,2235294,10919138,24000612,
%T 3074210315,6244442805,6244442805
%N Least k such that prime(k), prime(k+1), prime(k+2), ..., prime(k+n) all have the same last digit.
%C The interest in studying a sequence of n consecutive prime numbers having the same last digit is to look at the behavior of the rarefaction of these numbers when n becomes large.
%C a(k) > 10^10 for k >= 14. - _David A. Corneth_, Mar 22 2024
%e a(1) = A107730(1) = 34 because prime(34) = 139, prime(35) = 149, both end with the digit 9, and no two consecutive smaller primes end with the same digit.
%e a(2) = 258 because prime(258) = 1627, prime(259) = 1637, prime(260) = 1657 with the same last digit 7, and no three consecutive smaller primes have the same last digit.
%e a(4) = A371390(1).
%p nn:=15*10^6:
%p for n from 2 to 7 do :
%p ii:=0:d:=array(1..n):
%p for m from 1 to nn while(ii=0)
%p do:
%p lst:={}:
%p for k from 1 to n do:
%p d[k]:=irem(ithprime(m+k-1),10):
%p lst:=lst union {d[k]}:
%p od:
%p if lst={d[1]}
%p then
%p printf(`%d %d \n`,n-1,m):ii:=1:
%p else
%p fi:
%p od:
%p od:
%t a[n_] := Module[{v = Mod[Prime[Range[n + 1]], 10], k = 1, p}, p = Prime[n + 1]; While[! SameQ @@ v, p = NextPrime[p]; v = Join[Rest[v], {Mod[p, 10]}]; k++]; k]; Array[a, 6] (* _Amiram Eldar_, Mar 21 2024 *)
%o (PARI)
%o upto(n) = {
%o n += 30;
%o my(res = List(), q = 2, t = 1, ld = 2, nld, streak = 0);
%o forprime(p = 3, oo,
%o nld = p%10;
%o if(nld == ld,
%o streak++;
%o if(streak > #res,
%o listput(res, t-streak+1);
%o print1(t-streak+1", ");
%o )
%o ,
%o streak = 0
%o );
%o q = p;
%o ld = nld;
%o t++;
%o if(t > n,
%o return(res);
%o )
%o );
%o res
%o } \\ _David A. Corneth_, Mar 23 2024
%Y Cf. A000040, A107730, A129750, A371390.
%K nonn,base,hard,more
%O 1,1
%A _Michel Lagneau_, Mar 21 2024
%E a(7)-a(10) from _Amiram Eldar_, Mar 21 2024
%E a(11)-a(13) from _David A. Corneth_, Mar 22 2024