A107730 -id:A107730 - cp's OEIS frontend

A320703 Indices of primes followed by a gap (distance to next larger prime) of 10.

34, 42, 53, 61, 68, 80, 82, 101, 106, 115, 125, 127, 138, 141, 145, 157, 172, 175, 177, 191, 193, 204, 222, 233, 258, 266, 269, 279, 289, 306, 308, 310, 316, 324, 369, 383, 397, 399, 403, 418, 422, 431, 443, 474, 491, 497, 500, 502, 518, 525, 531, 535, 575

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031928.

Vincenzo Librandi, Table of n, a(n) for n = 1..8870
Index entries for primes, gaps between.

Equals A000720 o A031928.
Row 5 of A174349.
Indices of 10's in A001223.
Subsequence of A107730: prime(n+1) ends in same digit as prime(n).
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1000] | NthPrime(n+1) - NthPrime(n) eq 10]; // Vincenzo Librandi, Mar 21 2019

Select[Range[1000], Prime[#] + 10 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 21 2019 *)
Position[Partition[Prime[Range[600]],2,1],?(#[[2]]-#[[1]]==10&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Mar 08 2020 *)

A320703_vec(N=100,g=10,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031928(n)).

A320703 = { i > 0 | prime(i+1) = prime(i) + 10 }.

A320708 Indices of primes followed by a gap (distance to next larger prime) of 20.

Original entry on oeis.org

154, 259, 442, 480, 548, 753, 777, 783, 876, 971, 1035, 1066, 1095, 1106, 1147, 1254, 1277, 1302, 1337, 1345, 1355, 1381, 1396, 1400, 1423, 1438, 1562, 1592, 1613, 1662, 1669, 1808, 1955, 2016, 2043, 2081, 2116, 2129, 2147, 2226, 2302, 2307, 2387, 2517, 2547, 2563, 2694, 2724, 2745, 2755, 2766

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031938.

Vincenzo Librandi, Table of n, a(n) for n = 1..10720
Index entries for primes, gaps between.

Equals A000720 o A031938.
Row 10 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Indices of 20's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..3000] | NthPrime(n+1) - NthPrime(n) eq 20]; // Vincenzo Librandi, Mar 22 2019

Select[Range[3000], Prime[#] + 20 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)

A(N=100,g=20,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031938(n)).

A320708 = { i > 0 | prime(i+1) = prime(i) + 20 } = A001223^(-1)({20}).

A320713 Indices of primes followed by a gap (distance to next larger prime) of 30.

Original entry on oeis.org

590, 650, 708, 757, 842, 890, 928, 985, 1006, 1051, 1108, 1556, 1570, 1648, 1650, 1675, 1754, 1900, 1919, 2027, 2125, 2149, 2321, 2391, 2397, 2429, 2631, 2637, 2699, 2781, 2866, 2918, 2989, 2993, 3010, 3085, 3153, 3207, 3315, 3340, 3350, 3373, 3420, 3511, 3551, 3580, 3637, 3751, 3777, 3948

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A124596.

Index entries for primes, gaps between.

Equals A000720 o A124596.
Indices of 30's in A001223.
Row 15 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=30,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A124596(n)).

A320713 = { i>0 | prime(i+1) = prime(i) + 30 } = A001223^(-1)({30}).

A320718 Indices of primes followed by a gap (distance to next larger prime) of 40.

Original entry on oeis.org

2191, 2344, 2524, 2788, 3562, 4058, 4677, 5030, 5349, 6076, 6145, 6256, 6320, 6442, 6454, 6902, 7232, 7488, 8119, 8152, 8245, 8366, 8553, 8567, 8591, 8746, 9260, 9361, 10536, 10735, 11095, 11407, 11534, 11781, 12227, 12312, 12663, 12815, 12940, 13015, 13333, 13676, 13873, 14065, 14123

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A126721.

Index entries for primes, gaps between.

Equals A000720 o A126721.
Row 20 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Indices of 40's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=40,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A126721(n)).

A320718 = { i > 0 | prime(i+1) = prime(i) + 40 } = A001223^(-1)({40}).

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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A320703 Indices of primes followed by a gap (distance to next larger prime) of 10.

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A320708 Indices of primes followed by a gap (distance to next larger prime) of 20.

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A320713 Indices of primes followed by a gap (distance to next larger prime) of 30.

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A320718 Indices of primes followed by a gap (distance to next larger prime) of 40.

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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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Maple

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