A129750 -id:A129750 - cp's OEIS frontend

A107730 Numbers n such that prime(n+1) has the same last digit as prime(n).

34, 42, 53, 61, 68, 80, 82, 101, 106, 115, 125, 127, 138, 141, 145, 154, 157, 172, 175, 177, 191, 193, 204, 222, 233, 258, 259, 266, 269, 279, 289, 306, 308, 310, 316, 324, 369, 383, 397, 399, 403, 418, 422, 431, 442, 443, 474, 480, 491, 497, 500, 502, 518

Offset: 1

Jonathan Vos Post, Jun 12 2007

			a(1) = 34 because prime(34) = 139, prime(35) = 149, both end with the digit 9.
a(2) = 42 because prime(42) = 181, prime(43) = 191, both end with the digit 1.
a(4) = 61 because prime(61) = 283, prime(62) = 293, both end with the digit 3.
a(5) = 68 because prime(68) = 337, prime(69) = 347, both end with the digit 7.

Muniru A Asiru, Table of n, a(n) for n = 1..10000
Fred B. Holt, On the last digits of consecutive primes, arXiv:1604.02443 [math.NT], 2016.
Robert J. Lemke Oliver, Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
Index entries for primes, gaps between

Cf. A000040, A129750.

Union of rows r == 0 (mod 5) of A174349. Indices of multiples of 10 (A008592) in A001223.

P:=List(Filtered([1..4000],IsPrime),n->Reversed(ListOfDigits(n)));;
a:=Filtered([1..Length(P)-1],i->P[i+1][1]=P[i][1]); # Muniru A Asiru, Oct 31 2018

isA107730 := proc(n) local ldign, ldign2 ; ldign := convert(ithprime(n),base,10) ; ldign2 := convert(ithprime(n+1),base,10) ; if op(1,ldign) = op(1,ldign2) then true ; else false ; fi ; end: for n from 1 to 600 do if isA107730(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jun 15 2007

Select[Range[200],IntegerDigits[Prime[ # ]][[ -1]]==IntegerDigits[Prime[ #+1]][[ -1]]&] (* Stefan Steinerberger, Jun 14 2007 *)
Flatten[Position[Partition[Prime[Range[600]],2,1],?(Mod[#[[1]],10] == Mod[#[[2]],10]&),{1},Heads->False]] (* _Harvey P. Dale, Aug 20 2015 *)

isok(n) = (prime(n) % 10) == prime(n+1) % 10; \\ Michel Marcus, Feb 16 2017

is_A107730(n)=!((nextprime(1+n=prime(n))-n)%10) \\ This (...) is twice as fast as prime(n+1)-prime(n), and prime(n) becomes very slow for n > 41538, even with primelimit = 10^7. - M. F. Hasler, Oct 24 2018

Numbers n such that A000040(n)==A000040(n+1) mod 10, or A000040(n+1) - A000040(n) = 10*k for some integer k, or n such that A129750(n) = 0. [Corrected and edited by M. F. Hasler, Oct 24 2018]
A107730 = A001223^(-1)(A008592) = { i > 0 | A001223(i) == 0 (mod 10)} = U_{k>0} {A174349(5k,j); j >= 1}. - M. F. Hasler, Oct 24 2018
Union of A320703, A320708, A320713, A320718, ... A116493,..., A116496 ... etc. - R. J. Mathar, Apr 30 2024

More terms from Stefan Steinerberger and R. J. Mathar, Jun 14 2007

A371390 Numbers k such that prime(k), prime(k+1), prime(k+2), prime(k+3) and prime(k+4) all have the same last digit.

Original entry on oeis.org

11582, 17385, 19317, 20579, 22931, 42098, 51895, 52252, 55259, 60393, 62192, 62193, 62680, 64050, 65324, 71483, 76391, 76773, 76805, 77052, 81139, 86711, 95661, 100208, 102032, 113646, 113892, 113954, 115251, 124227, 125218, 125586, 144165, 144299, 147619, 147620

Offset: 1

Michel Lagneau, Mar 20 2024

			11582 is a term because prime(11582) = 123229, prime(11583) = 123239, prime(11584) = 123259, prime(11585) = 123269 with the same last digit 9.

Robert Israel, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

Cf. A000040, A007652, A107730, A129750.

nn:=15*10^4:d:=array(1..5):
for n from 1 to nn do:
 for k from 1 to 5 do:
   d[k]:=irem(ithprime(n+k-1),10):
 od:
  if d[1]=d[2] and d[1]=d[3] and
d[1]=d[4] and d[1]=d[5]
    then
     printf(`%d, `,n):
    else
  fi:
od:

PARI
```
\\ See PARI link
```

from itertools import count, islice
from sympy import nextprime
def A371390_gen(): # generator of terms
    xlist, p = [2, 3, 5, 7, 1], 11
    for k in count(1):
        if len(set(xlist)) == 1:
            yield k
        p = nextprime(p)
        xlist = xlist[1:]+[p%10]
A371390_list = list(islice(A371390_gen(),10)) # Chai Wah Wu, Apr 13 2024

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

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Mar 21 2024

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

			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).

Cf. A000040, A107730, A129750, A371390.

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:

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 *)

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

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

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A107730 Numbers n such that prime(n+1) has the same last digit as prime(n).

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A371390 Numbers k such that prime(k), prime(k+1), prime(k+2), prime(k+3) and prime(k+4) all have the same last digit.

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

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