A124596 -id:A124596 - cp's OEIS frontend

A052195 Primes p such that p, p+30, p+60 are consecutive primes.

69593, 110651, 134609, 228647, 237791, 250889, 303157, 318919, 396449, 421913, 498271, 507431, 535243, 554317, 629623, 642427, 642457, 668243, 692161, 716003, 729791, 780523, 782581, 790897, 801217, 825131, 829289, 847393, 892291, 902873, 940097, 942449, 963913, 995243, 1027067

Offset: 1

Labos Elemer, Jan 28 2000

nonn

			69593, 69623, 69653 are consecutive primes with equal distance d = 30.
110651, 110681 and 110711 are consecutive primes with equal distance d = 30.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020.

Subsequence of A124596 (primes followed by gap 30).
Cf. A047948 (analog for gap 6), A052188 (gap 12), A052189 (gap 18), A052190 (gap 24), A053075 (a(n) + 30).
Cf. A001223 (gaps), A052243 (quadruplets with gap 30), A033451 (quadruplets with gap 6).

Select[Partition[Prime[Range[80000]],3,1],Differences[#]=={30,30}&][[All,1]] (* Harvey P. Dale, May 03 2018 *)

vecextract(A124596, select(t->t==30, A124596[^1]-A124596[^-1],1)) \\ Terms of A124596 with indices of first differences of 30. Gives a(1..230) from A124596(1..10^4). - M. F. Hasler, Jan 02 2020

{ A124596(n) | A124596(n+1) = A124596(n) + 30 }. - M. F. Hasler, Jan 02 2020

A126784 Primes p such that q-p = 32, where q is the next prime after p.

Original entry on oeis.org

5591, 10799, 27701, 27851, 33647, 39047, 41081, 41687, 43721, 44417, 45989, 47459, 50789, 52457, 55259, 55547, 61781, 62351, 64817, 66239, 67307, 69959, 73907, 79907, 80567, 82307, 84089, 88037, 94169, 94961, 99191, 99929, 100559, 102611

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Feb 24 2007

Lower prime of a difference of 32 between consecutive primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001359, A029710, A031924, A031926, A031928, A031930, A031932, A031934, A031936, A031938, A061779, A098974, A124594, A124595, A124596, A000230, A050434.

lista(nn) = {p = 2; while (p < nn, q = nextprime(p+1); if (q - p == 32, print1(p, ", ")); p = q;);} \\ Michel Marcus, Jul 17 2013

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

A053075 Primes p such that p-30, p, p+30 are consecutive primes.

Original entry on oeis.org

69623, 110681, 134639, 228677, 237821, 250919, 303187, 318949, 396479, 421943, 498301, 507461, 535273, 554347, 629653, 642457, 642487, 668273, 692191, 716033, 729821, 780553, 782611, 790927, 801247, 825161, 829319, 847423, 892321, 902903

Offset: 1

Harvey P. Dale, Feb 25 2000

Original name: Primes p(k) such that p(k) - p(k-1) = p(k+1) - p(k) = 30.

			110681 is separated from both the next lower prime and the next higher prime by 30

Zak Seidov, Table of n, a (n) for n = 1 .. 1000

Cf. A052195 (a(n)-30: first of the triplets) and cross-references there.

Subsequence of A124596 (primes followed by gap 30).

lst={}; Do[p=Prime[n]; If[p-Prime[n-1] == Prime[n+1]-p == 6*5, AppendTo[lst,p]], {n,2,2*8!}]; lst (* Vladimir Joseph Stephan Orlovsky, May 20 2010 *)

is_A053075(n)={precprime(n-1)==n-30&&nextprime(n+1)==n+30&&isprime(n)} \\ M. F. Hasler, Jan 02 2020

a(n) = A052195(n) + 30. - Zak Seidov, Dec 21 2012

A052195 = { A124596(n) | A124596(n-1) = A124596(n) - 30 }. - M. F. Hasler, Jan 02 2020

Name edited to conform with style sheet and A052195 etc. - M. F. Hasler, Jan 02 2020

A204672 Primes followed by a gap of 120.

Original entry on oeis.org

1895359, 2898239, 6085441, 7160227, 7784039, 7803491, 7826899, 8367397, 8648557, 9452959, 10052071, 10863973, 11630503, 11962823, 12109697, 12230233, 12415681, 14411737, 14531899, 15014557, 15020737, 15611909, 16179041

Offset: 1

M. F. Hasler, Jan 18 2012

nonn

M. F. Hasler, Table of n, a(n) for n = 1..433 (all a(n) < 10^8).
Index entries for primes, gaps between

Cf. A058193 (first gap of 6n), A140791 (first gap of 10n).
Cf. A126771 (gap 60), A126724 (gap 150), A204673 (gap 180).
Cf. A001359, A029710, A031924-A031938, A061779, A098974, A124594-A124596, A126784, A134116-A134124, A204665-A204670.

N = 2*10^7; % to get all terms <= N
P = primes(N+120);
J = find(P(2:end) - P(1:end-1) == 120);
P(J)  % Robert Israel, Feb 28 2017

Transpose[Select[Partition[Prime[Range[1100000]],2,1],Last[#]-First[#] == 120&]] [[1]] (* Harvey P. Dale, Jul 11 2014 *)

g=120;c=o=0;forprime(p=1,default(primelimit),(-o+o=p)==g&write("c:/temp/b204672.txt",c++" "p-g))

A174350 Square array: row n >= 1 lists the primes p for which the next prime is p+2n; read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 16 2010

Every odd prime p = prime(i), i > 1, occurs in this array, in row (prime(i+1) - prime(i))/2. Polignac's conjecture states that each row contains an infinite number of indices. In case this does not hold, we can use the convention to continue finite rows with 0's, to ensure the sequence is well defined. - M. F. Hasler, Oct 19 2018

A permutation of the odd primes (A065091). - Robert G. Wilson v, Sep 13 2022

			Upper left hand corner of the array:
     3     5    11    17    29    41    59    71   101 ...
     7    13    19    37    43    67    79    97   103 ...
    23    31    47    53    61    73    83   131   151 ...
    89   359   389   401   449   479   491   683   701 ...
   139   181   241   283   337   409   421   547   577 ...
   199   211   467   509   619   661   797   997  1201 ...
   113   293   317   773   839   863   953  1409  1583 ...
  1831  1933  2113  2221  2251  2593  2803  3121  3373 ...
   523  1069  1259  1381  1759  1913  2161  2503  2861 ...
  (...)
Row 1: p(2) = 3, p(3) = 5, p(5) = 11, p(7) = 17,... these being the primes for which the next prime is 2 greater: (lesser of) twin primes A001359.
Row 2: p(4) = 7, p(6) = 13, p(8) = 19,... these being the primes for which the next prime is 4 greater: (lesser of) cousin primes A029710.

Robert G. Wilson v, Falling antidiagonals 1..115, flattened (first 50 from T. D. Noe).
Fred B. Holt and Helgi Rudd, On Polignac's Conjecture, arxiv:1402.1970 [math.NT], 2014.

Cf. A000040, A001223, A065091, A174349.
Rows 1, 2, 3, ...: A001359, A029710, A031924, A031926, A031928 (row 5), A031930, A031932, A031934, A031936, A031938 (row 10), A061779, A098974, A124594, A124595, A124596 (row 15), A126784, A134116, A134117, A134118, A126721 (row 20), A134120, A134121, A134122, A134123, A134124 (row 25), A204665, A204666, A204667, A204668, A126771 (row 30), A204669, A204670.
Rows 35, 40, 45, 50, ...: A204792, A126722, A204764, A050434 (row 50), A204801, A204672, A204802, A204803, A126724 (row 75), A184984, A204805, A204673, A204806, A204807 (row 100); A224472 (row 150).
Column 1: A000230.
Column 2: A046789.

rows = 10; t2 = {}; Do[t = {}; p = Prime[2]; While[Length[t] < rows - off + 1, nextP = NextPrime[p]; If[nextP - p == 2*off, AppendTo[t, p]]; p = nextP]; AppendTo[t2, t], {off, rows}]; Table[t2[[b, a - b + 1]], {a, rows}, {b, a}] (* T. D. Noe, Feb 11 2014 *)
t[r_, 0] = 2; t[r_, c_] := Block[{p = NextPrime@ t[r, c - 1], q}, q = NextPrime@ p; While[ p + 2r != q, p = q; q = NextPrime@ q]; p]; Table[ t[r - c + 1, c], {r, 10}, {c, r, 1, -1}] (* Robert G. Wilson v, Nov 06 2020 *)

A174350_row(g, N=50, i=0, p=prime(i+1), L=[])={g*=2; forprime(q=1+p, , i++; if(p+g==p=q, L=concat(L, q-g); N--||return(L)))} \\ Returns the first N terms of row g. - M. F. Hasler, Oct 19 2018

a(n) = A000040(A174349(n)). - Michel Marcus, Mar 30 2016

Definition corrected and other edits by M. F. Hasler, Oct 19 2018

A224472 Primes followed by a gap of 300.

Original entry on oeis.org

4758958741, 5612345261, 6169169561, 6306815239, 6646984159, 7335508261, 8645089003, 8806019249, 9047808247, 9148138313, 9466071347, 9907846261, 10055451683, 11063821453, 11475026363, 11603081459, 12292390637, 12750876857, 13833827471, 14636472007, 15876700949

Offset: 1

Zak Seidov, Apr 07 2013

nonn

The first twin gap equal to 300 occurs for p = 6537587646371. - Giovanni Resta, Apr 07 2013

Zak Seidov, Table of n, a(n) for n = 1..3718
Index entries for primes, gaps between

Cf. A058193 (first gap of 6n), A140791 (first gap of 10n), A126771 (gap 60), A126724 (gap 150), A204673 (gap 180), A204807 (gap 200), A000230, A001359, A204672, A029710, A031924-A031938, A061779, A098974, A124594-A124596, A126784, A134116-A134124, A204665-A204670.

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}).

A270754 Numbers n such that n - 31, n - 1, n + 1 and n + 31 are consecutive primes.

Original entry on oeis.org

90438, 258918, 293862, 385740, 426162, 532950, 1073952, 1317192, 1318410, 1401318, 1565382, 1894338, 1986168, 2174772, 2612790, 2764788, 3390900, 3450048, 3618960, 3797250, 3961722, 3973062, 4074870, 4306230, 4648068, 4917360, 5351010, 5460492

Offset: 1

Karl V. Keller, Jr., Mar 22 2016

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 are divisible by 30 (cf. A249674).
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.
The numbers n - 31 and n + 1 belong to A049481 (p and p + 30 are primes) and A124596 (p where p + 30 is the next prime).
The numbers n - 31 and n - 1 belong to A049489 (p and p + 32 are primes).

			90438 is the average of the four consecutive primes 90407, 90437, 90439, 90469.
258918 is the average of the four consecutive primes 258887, 258917, 258919, 258949.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

from sympy import isprime,prevprime,nextprime
for i in range(0,1000001,6):
   if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-31 and nextprime(i+1) == i+31 :  print (i,end=', ')

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A052195 Primes p such that p, p+30, p+60 are consecutive primes.

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A126784 Primes p such that q-p = 32, where q is the next prime after p.

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

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A053075 Primes p such that p-30, p, p+30 are consecutive primes.

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A204672 Primes followed by a gap of 120.

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A174350 Square array: row n >= 1 lists the primes p for which the next prime is p+2n; read by antidiagonals.

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A224472 Primes followed by a gap of 300.

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

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