A371403 Least k such that prime(k), prime(k+1), prime(k+2), ..., prime(k+n) all have the same last digit.

34, 258, 2147, 11582, 62192, 274810, 1500309, 2235294, 10919138, 24000612, 3074210315, 6244442805, 6244442805, 143338476264, 244844614858

Offset: 1

Author

Michel Lagneau, Mar 21 2024

Keywords

• nonn
• base
• hard
• more

Comments

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.

a(k) > 10^10 for k >= 14. - David A. Corneth, Mar 22 2024

Examples

			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.
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.
a(4) = A371390(1).
		

Crossrefs

Cf. A000040, A107730, A129750, A371390.

Programs

Maple

nn:=15*10^6:
for n from 2 to 7 do :
   ii:=0:d:=array(1..n):
  for m from 1 to nn while(ii=0)
do:
   lst:={}:
     for k from 1 to n do:
d[k]:=irem(ithprime(m+k-1),10):
        lst:=lst union {d[k]}:
     od:
      if lst={d[1]}
       then
       printf(`%d %d \n`,n-1,m):ii:=1:
       else
      fi:
    od:
    od:

Mathematica

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 *)

PARI

upto(n) = {
	n += 30;
	my(res = List(), q = 2, t = 1, ld = 2, nld, streak = 0);
	forprime(p = 3, oo,
		nld = p%10;
		if(nld == ld,
			streak++;
			if(streak > #res,
				listput(res, t-streak+1);
				print1(t-streak+1", ");
			)
		,
			streak = 0
		);
		q = p;
		ld = nld;
		t++;
		if(t > n,
			return(res);
		)
	);
	res
} \\ David A. Corneth, Mar 23 2024

Extensions

a(7)-a(10) from Amiram Eldar, Mar 21 2024

a(11)-a(13) from David A. Corneth, Mar 22 2024

a(14) from Michael S. Branicky, May 15 2025

a(15) from Michael S. Branicky, May 21 2025