A031928 -id:A031928 - cp's OEIS frontend

A160500 Prime p1 of consecutive primes p1, p2, where p2-p1=10, and p1, p2 are in different centuries.

691, 1399, 1699, 5791, 6091, 6691, 6793, 7297, 8599, 10993, 12391, 12799, 13999, 14197, 14293, 15091, 15391, 15991, 17599, 18493, 18691, 19699, 22699, 22993, 23899, 24499, 24799, 25693, 26893, 27397, 28099, 28297, 28393, 29191, 33493

Offset: 1

Ki Punches, May 15 2009

nonn

The sequence is probably infinite.

It appears that every integer occurs as the difference round((a(n+1)-a(n))/100); all numbers 1..228 occur as these differences for a(n) < 1000000000. - Hartmut F. W. Hoft, May 18 2017

			Consecutive primes 10993 and 11003 differ by 10 and are in consecutive centuries, so 10993 is in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A160440, A160370, A031928.

a160500[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[100, n, 100]], First[#]-Last[#]==10&]]
a160500[33500] (* data *) (* Hartmut F. W. Hoft, May 18 2017 *)
cpdcQ[{a_,b_}]:=b-a==10&&Floor[a/100]!=Floor[b/100]; Select[Partition[Prime[Range[ 4000]],2,1],cpdcQ][[;;,1]] (* Harvey P. Dale, Sep 02 2024 *)

Edited by Ray Chandler, May 22 2009

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

Original entry on oeis.org

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 }.

A052376 Primes followed by a [10,2,10] prime difference pattern of A001223.

Original entry on oeis.org

409, 1039, 2017, 2719, 3571, 4219, 4231, 4261, 4327, 6079, 6121, 6679, 6781, 8209, 11047, 11149, 11959, 12241, 15277, 19531, 19687, 21577, 21589, 26881, 27529, 28087, 28297, 29389, 30829, 30859, 31069, 32401, 42061, 45307, 47797, 48109

Offset: 1

Labos Elemer, Mar 22 2000

nonn

Subsequence of lesser terms of 10-twins (A031928).
Start primes of quadruples consisting of two consecutive 10-twins of prime which are in minimal distance [minD = A052380(10/2) = 12].
First term of this sequence is 409 = A052381(5).

			p=1039 begins [1039,1049,1051,1061] prime quadruple with the appropriate difference pattern: [10,2,10]=[d,D-d,d], so d=10, D=12.

Robert G. Wilson v, Table of n, a(n) for n = 1..1780

Cf. A031928, A053323, A052380, A052381, A052377, A052378.

{p, q, r, s} = {2, 3, 5, 7}; lst = {}; While[p < 50000, If[ Differences[{p, q, r, s}] == {10, 2, 10}, AppendTo[lst, p]]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; lst (* Robert G. Wilson v, Jul 15 2015 *)

a(n)=p, a prime which begins a [p, p+d, p+D, p+D+d]=[p, p+10, p+12, p+22] prime quadruple.

a(n) = A259025(n)-11. - Robert G. Wilson v, Jul 15 2015

A192175 Array of primes determined by distance to next prime, by antidiagonals.

Original entry on oeis.org

2, 3, 7, 5, 13, 23, 11, 19, 31, 89, 17, 37, 47, 359, 139, 29, 43, 53, 389, 181, 199, 41, 67, 61, 401, 241, 211, 113, 59, 79, 73, 449, 283, 467, 293, 1831, 71, 97, 83, 479, 337, 509, 317, 1933, 523, 101, 103, 131, 491, 409, 619, 773, 2113, 1069, 887, 107

Offset: 1

Clark Kimberling, Jun 24 2011

Row 1:  primes p such that p+1 or p+2 is a prime.
Row r>1:  primes p such that the least h for which p+2h is prime is r.
Rows 1-7:  A124588, A023200, A031924, A031926, A031928, A031932, A031924.

			Northwest corner:
  2.....3.....5.....11....17....29....41
  7.....13....19....37....43....67....79
  23....31....47....53....61....73....83
  89....359...389...401...449...479...491
  139...181...241...283...337...409...421
For example, 31 is in row 3 because 31+2*3 is a prime, unlike 31+2*1 and 31+2*2.  Every prime occurs exactly once.  For each row, it is not known whether it is finite.

Cf. A192176, A192177, A192178, A192179.

z = 5000; (* z=number of primes used *)
row[1] = (#1[[1]] &) /@ Cases[Array[{#1,
      PrimeQ[1 + Prime[#1]] || PrimeQ[2 + Prime[#1]]} &, {z}], {_, True}];
Do[row[x] = Complement[(#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[2 x + Prime[#1]]} &, {z}], {_, True}], Flatten[Array[row, {x - 1}]]], {x, 2, 16}]; TableForm[Array[Prime[row[#]] &, {10}]] (* A192175 array *)
Flatten[Table[ Prime[row[k][[n - k + 1]]], {n, 1, 11}, {k, 1, n}]] (* A192175 sequence *)
(* Peter J. C. Moses, Jun 20 2011 *)

A290450 Primes with property that the next prime has the same last digit.

Original entry on oeis.org

139, 181, 241, 283, 337, 409, 421, 547, 577, 631, 691, 709, 787, 811, 829, 887, 919, 1021, 1039, 1051, 1153, 1171, 1249, 1399, 1471, 1627, 1637, 1699, 1723, 1801, 1879, 2017, 2029, 2053, 2089, 2143, 2521, 2647, 2719, 2731, 2767, 2887, 2917, 3001, 3089, 3109, 3361, 3413, 3517, 3547, 3571

Offset: 1

Alonso del Arte, Aug 06 2017

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

			139 is in the sequence because the immediately following prime is 149, which also ends in 9.
But 149 is not in the sequence because the next prime after that one is 151, which ends in 1, not 9.

Marius A. Burtea, Table of n, a(n) for n = 1..5005
Evelyn Lamb, "Peculiar Pattern Found in 'Random' Prime Numbers", Nature, March 14, 2016, republished by Scientific American.
Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv preprint arXiv:1603.03720 [math.NT], submitted on March 11, 2016.

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

f:=func; 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

Select[Partition[Prime[Range[1000]], 2, 1], Mod[#[[1]], 10] == Mod[#[[2]], 10] &][[All, 1]] (* Harvey P. Dale, Aug 21 2017 *)
Module[{nn=1000,prs,p},prs=Prime[Range[nn]];p=Divisible[#,10]&/@ Differences[prs];Pick[Most[prs],p]] (* Harvey P. Dale, Aug 22 2017 *)

A031928 UNION A031938 UNION A124596 UNION A126721 UNION ... - R. J. Mathar, Jan 23 2022

A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

Original entry on oeis.org

7, 19, 37, 67, 79, 97, 109, 127, 229, 277, 307, 349, 379, 397, 439, 457, 487, 499, 739, 757, 769, 859, 877, 907, 937, 967, 1009, 1087, 1279, 1297, 1429, 1447, 1489, 1549, 1567, 1579, 1597, 1609, 1867, 1999, 2137, 2239, 2269, 2347, 2377, 2389, 2437, 2539, 2617, 2659, 2689, 2707, 2749, 2797, 2857

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's or 9's.

			7 is in this sequence since pair (7,11) is the first with difference 4 spanning a multiple of 10.

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288022, A288024.

a288021[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==4&]]
a288021[3000] (* data *)

A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

Original entry on oeis.org

47, 157, 167, 257, 367, 557, 587, 607, 647, 677, 727, 947, 977, 1097, 1117, 1187, 1217, 1367, 1657, 1747, 1777, 1907, 1987, 2207, 2287, 2417, 2467, 2677, 2837, 2897, 2957, 3307, 3407, 3607, 3617, 3637, 3727, 3797, 4007, 4357, 4457, 4507, 4597, 4657, 4937, 4987

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 7's.

Number of terms < 10^k: 0, 0, 1, 13, 81, 565, 4027, 30422, 237715, ... - Muniru A Asiru, Jan 09 2018

			47 is in the sequence since pair (47,53) is the first with difference 6 spanning a multiple of 10.

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288024.

P:=Filtered([1..20000], IsPrime);
P1:=List(Filtered(Filtered(List([1..Length(P)-1],n->[P[n],P[n+1]]),i->i[2]-i[1]=6),j->j[1] mod 5=2),k->k[1]); # Muniru A Asiru, Jul 08 2017

for n from 1 to 2000 do if [ithprime(n+1)-ithprime(n), ithprime(n) mod 5] = [6,2] then print(ithprime(n)); fi; od; # Muniru A Asiru, Jan 19 2018

a288022[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==6&]]
a288022[3000] (* data *)

A288024 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 8, and p1, p2 are in different decades.

Original entry on oeis.org

89, 359, 389, 449, 479, 683, 719, 743, 929, 983, 1109, 1163, 1193, 1373, 1439, 1523, 1559, 1733, 1823, 1979, 2003, 2153, 2213, 2243, 2273, 2459, 2609, 2663, 2699, 2843, 2879, 2909, 3209, 3449, 3623, 3719, 4289, 4349, 4583, 4943, 5189, 5399, 5573, 5693, 5783, 5813

Offset: 1

Hartmut F. W. Hoft, Jun 04 2017

nonn

The unit digits of the numbers in the sequence are 3's or 9's.

			89 is in the sequence since pair (89,97) is the first with difference 8 spanning a multiple of 10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049, A287050, A060229, A288021, A288022.

a288024[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[10, n, 10]], First[#]-Last[#]==8&]]
a288024[6000] (* data *)
Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==8&&IntegerDigits[#[[1]]][[-2]]!= IntegerDigits[ #[[2]]][[-2]]&][[;;,1]] (* Harvey P. Dale, Jan 09 2024 *)

A286891 Initial primes of 6 consecutive primes with 5 consecutive gaps 10, 8, 6, 4, 2.

Original entry on oeis.org

41203, 556243, 576193, 715849, 752263, 859249, 891799, 1107763, 1191079, 1201999, 1210369, 1510189, 1601599, 1893163, 2530963, 2678719, 2881243, 3257689, 3431479, 3545263, 3792949, 3804919, 4041109, 4479463, 4867309

Offset: 1

Muniru A Asiru, Jul 22 2017

nonn

All terms = {13,19} mod 30.

For initial primes of 6 consecutive primes with consecutive gaps 2, 4, 6, 8, 10 see A190817.

			Prime(4313..4318) = {41203, 41213, 41221, 41227, 41231, 41233} and 41203 + 10 = 41213, 41213 + 8 = 41221, 41221 + 6 = 41227, 41227 + 4 = 41231, 41231 + 2 = 41233.
Also, prime(68287..68292) = {859249, 859259, 859267, 859273, 859277, 859279} and 859249 + 10 = 859259, 859259 + 8 = 859267, 859267 + 6 = 859273, 859273 + 4 = 859277, 859277 + 2 = 859279.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
R. J. Mathar, Table of Prime Gap Constellations

Cf. A078847, A190814, A190817, A190819, A190838, A290161, A290162.

Subsequence of A031928.

P:=Filtered([1..20000000],IsPrime);;  I:=Reversed([2,4,6,8,10]);;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4]]);;
P3:=List(Positions(P2,I),i->P[i]);

K:=10^7: # to get all terms <= K.
Primes:=select(isprime,[seq(i,i=3..K+30,2)]): Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1], Primes[t+3]-Primes[t+2], Primes[t+4]-Primes[t+3], Primes[t+5]-Primes[t+4]]=[10,8,6,4,2], [$1..nops(Primes)-5])]; # Muniru A Asiru, Aug 15 2017

Select[Partition[Prime[Range[340000]],6,1],Differences[#]=={10,8,6,4,2}&][[All,1]] (* Harvey P. Dale, Aug 22 2018 *)

A287050 Square array read by antidiagonals upwards: M(n,k) is the initial occurrence of first prime p1 of consecutive primes p1, p2, where p2 - p1 = 2*k, and p1, p2 span a multiple of 10^n, n>=1, k>=1.

Original entry on oeis.org

29, 599, 7, 2999, 97, 47, 179999, 1999, 1097, 89, 23999999, 69997, 21997, 1193, 139, 23999999, 199999, 369997, 23993, 691, 199, 29999999, 19999999, 3199997, 149993, 10993, 199, 113, 17399999999, 19999999, 6999997, 1199999, 139999, 997, 293, 1831

Offset: 1

Hartmut F. W. Hoft, May 18 2017

The unit digits of the numbers in the matrix representation M(n,k) are 9's for column 1, 7's or 9's for column 2, 7's for column 3, 3's or 9's for column 4, and 1's, 3's, 7's or 9's for column 5.
The following matrix terms appear as first terms in sequence
A060229(1) = M(1,1)
A288021(1) = M(1,2)
A288022(1) = M(1,3)
A288024(1) = M(1,4)
A031928(1) = M(1,5)
A158277(1) = M(2,1)
A160440(1) = M(2,2)
A160370(1) = M(2,3)
A287049(1) = M(2,4)
A160500(1) = M(2,5)
A158861(1) = M(3,1).

			The matrix representation of the sequence with row n indicating the spanned power of 10 and column k indicating the difference of 2*k between the first pair of consecutive primes spanning a multiple of 10^n:
--------------------------------------------------------------------------
n\k   1             2             3             4            5
--------------------------------------------------------------------------
1 |   29            7             47            89           139
2 |   599           97            1097          1193         691
3 |   2999          1999          21997         23993        10993
4 |   179999        69997         369997        149993       139999
5 |   23999999      199999        3199997       1199999      1999993
6 |   23999999      19999999      6999997       38999993     1999993
7 |   29999999      19999999      159999997     659999999    379999999
8 |   17399999999   7699999999    9399999997    8999999993   499999993
9 |   92999999999   135999999997  85999999997   8999999993   28999999999
10|   569999999999  519999999997  369999999997  29999999993  819999999997
...
Every column in the matrix is nondecreasing.
For the first and fourth columns, ceiling(M[n,1]/10^n) and ceiling(M[n,4]/10^n) are divisible by 3, for all n>=1 (see A158277 and A287049).

Cf. A001359, A023201, A023203, A031924, A031925, A031928, A046117, A046132, A158277, A158861, A160370, A160440, A160500, A287049.

Cf. A060229, A288021, A288022, A288024.

M(n,k) = min( p_i : p_(i+1) - p_i = 2*k, p_i and p_(i+1) consecutive primes and p_i < m*10^n < p_(i+1) for some integer m) where p_j is the j-th prime, n>=1 and k>=1.

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A160500 Prime p1 of consecutive primes p1, p2, where p2-p1=10, and p1, p2 are in different centuries.

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

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A052376 Primes followed by a [10,2,10] prime difference pattern of A001223.

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A192175 Array of primes determined by distance to next prime, by antidiagonals.

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A290450 Primes with property that the next prime has the same last digit.

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A288021 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 4, and p1, p2 are in different decades.

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A288022 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 6, and p1, p2 are in different decades.

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A288024 Prime p1 of consecutive primes p1, p2, where p2 - p1 = 8, and p1, p2 are in different decades.

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