%I #48 Mar 08 2024 01:15:28

%S 139,181,241,283,337,409,421,547,577,631,691,709,787,811,829,887,919,

%T 1021,1039,1051,1153,1171,1249,1399,1471,1627,1637,1699,1723,1801,

%U 1879,2017,2029,2053,2089,2143,2521,2647,2719,2731,2767,2887,2917,3001,3089,3109,3361,3413,3517,3547,3571

%N Primes with property that the next prime has the same last digit.

%C Starts off the same as A031928, primes p such that the next prime is p + 10. First term that differs is 887, since 897 = 3 * 13 * 23 and the next prime is 907.

%C As the primes get larger and more sparsely distributed, the difference between successive primes is less likely to be less than 10.

%C One might expect that a prime is 1/4 as likely to be followed by a prime with the same least significant digit in base 10 (since the possibilities are 1, 3, 7, 9).

%C One might also expect this sequence to consist of a quarter of the primes. And yet pi(a(50)) = pi(3547) = 497; the 200th prime is 1223.

%H Marius A. Burtea, <a href="/A290450/b290450.txt">Table of n, a(n) for n = 1..5005</a>

%H Evelyn Lamb, <a href="https://www.scientificamerican.com/article/peculiar-pattern-found-in-random-prime-numbers/">"Peculiar Pattern Found in 'Random' Prime Numbers"</a>, Nature, March 14, 2016, republished by Scientific American.

%H Robert J. Lemke Oliver and Kannan Soundararajan, <a href="https://arxiv.org/abs/1603.03720">Unexpected biases in the distribution of consecutive primes</a>, arXiv preprint arXiv:1603.03720 [math.NT], submitted on March 11, 2016.

%F A031928 UNION A031938 UNION A124596 UNION A126721 UNION ... - _R. J. Mathar_, Jan 23 2022

%e 139 is in the sequence because the immediately following prime is 149, which also ends in 9.

%e But 149 is not in the sequence because the next prime after that one is 151, which ends in 1, not 9.

%t Select[Partition[Prime[Range[500000]], 2, 1], Mod[#[[1]], 10] == Mod[#[[2]], 10] &][[All, 1]] ( * _Harvey P. Dale_, Aug 21 2017 *)

%t Module[{nn=500000,prs,p},prs=Prime[Range[nn]];p=Divisible[#,10]&/@ Differences[prs];Pick[Most[prs],p]] (* _Harvey P. Dale_, Aug 22 2017 *)

%o (Magma) f:=func<p,m|p mod 10 eq m and NextPrime(p) mod 10 eq m>; a:=[]; for p in PrimesUpTo(4000) do if f(p,1) or f(p,3) or f(p,7) or f(p,9) then Append(~a,p); end if; end for; a; // _Marius A. Burtea_, Oct 16 2019

%Y Cf. A031928 (subset), A050434 (with 2 digits).

%K nonn,base

%O 1,1

%A _Alonso del Arte_, Aug 06 2017