A031925 -id:A031925 - cp's OEIS frontend

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.

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.

A333200 Rectangular array read by antidiagonals: row n shows the primes p(k) such that p(k) = p(k-1) + 2n, with 2 prefixed to row 1.

Original entry on oeis.org

2, 3, 11, 5, 17, 29, 7, 23, 37, 97, 13, 41, 53, 367, 149, 19, 47, 59, 397, 191, 211, 31, 71, 67, 409, 251, 223, 127, 43, 83, 79, 457, 293, 479, 307, 1847, 61, 101, 89, 487, 347, 521, 331, 1949, 541, 73, 107, 137, 499, 419, 631, 787, 2129, 1087, 907, 103, 113

Offset: 1

Clark Kimberling, May 09 2020

Every prime occurs exactly once.
Row 1: A001632, except for initial term
Row 2: A046132
Row 3: A031925
Row 4: A031927
Row 5: A031929
Column 1: A006512, beginning with 5,7,13

			Northwest corner:
    2   3     5    7   13   19   31   43   61   73  103
   11   17   23   41   47   71   83  101  107  113  131
   29   37   53   59   67   79   89  137  157  163  173
   97  367  397  409  457  487  499  691  709  727  751
  149  191  251  293  347  419  431  557  587  641  701

Cf. A333201, A000040, A001632, A006512.

z = 2700; p = Prime[Range[z]];
r[n_] := Select[Range[z], p[[#]] - p[[# - 1]] == 2 n &]; r[1] = Join[{1, 2}, r[1]];
TableForm[Table[Prime[r[n]], {n, 1, 18}]]  (* A333200, array *)
TableForm[Table[r[n], {n, 1, 18}]] (* A333201, array *)
Table[Prime[r[n - k + 1][[k]]], {n, 12}, {k, n, 1, -1}] // Flatten (* A333200, sequence *)
Table[r[n - k + 1][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* A333201, sequence *)

A052159 Lower prime of a difference of 6 (G-minor-6 primes) between consecutive primes of 6k+5 form.

Original entry on oeis.org

23, 47, 53, 83, 131, 167, 173, 233, 251, 257, 263, 353, 383, 443, 503, 557, 563, 587, 593, 647, 653, 677, 941, 947, 971, 977, 1013, 1097, 1103, 1181, 1187, 1217, 1223, 1283, 1361, 1367, 1433, 1493, 1553, 1601, 1613, 1901, 1907, 1973, 2063, 2207, 2333

Offset: 1

Labos Elemer, Jan 25 2000

nonn

The corresponding larger primes (G-major-6 primes) are also of the form 6k+5.

			a(1)=23 since a(1) + 6 = 29 is the next prime and 29 = 6*4 + 5.

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

Cf. A031924, A031925.

Select[Partition[Prime[Range[400]],2,1],#[[2]]-#[[1]]==6&&Mod[#,6]=={5,5}&][[All,1]] (* Harvey P. Dale, Jan 05 2022 *)

A031924(n) == 5 (mod 6).

A052230 Primes p from A031924 such that A052180(primepi(p)) = 5.

Original entry on oeis.org

23, 31, 53, 61, 83, 151, 173, 233, 263, 271, 331, 353, 383, 443, 503, 541, 563, 571, 593, 601, 653, 751, 991, 1013, 1103, 1223, 1231, 1283, 1291, 1321, 1433, 1493, 1553, 1613, 1621, 1741, 1861, 1973, 2011, 2063, 2131, 2281, 2333, 2341, 2371, 2393, 2543

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A031924, A031925, A052180.

filter:= proc(p) local t,m,flag;
  flag:= false;
  for t from p+1 to p+5 do
    m:= min(numtheory:-factorset(t));
    if m > 5 then return false
    elif m = 5 then flag:= true
    fi
  od;
  flag
end proc:
Res:= NULL: count:= 0:
q:= 1: p:= 2:
while count < 100 do
  q:= p;
  p:= nextprime(p);
  if p-q = 6 and filter(q) then
    count:= count+1; Res:= Res, q;
  fi
od:
Res; # Robert Israel, Aug 12 2018

A126720 Primes p such that p - q = 24, where q is the previous prime before p; or prime numbers preceded by precisely 23 composite numbers.

Original entry on oeis.org

1693, 2203, 4201, 4547, 4783, 5261, 6197, 6421, 6761, 7103, 7393, 7817, 8147, 8353, 9091, 11027, 11657, 11863, 12097, 12143, 13033, 13291, 16057, 16217, 16477, 16787, 16811, 17077, 17707, 18013, 18617, 18661, 19207, 19531, 20507, 22433, 22901

Offset: 1

Artur Jasinski, Feb 13 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000230, A033560, A031505, A031925, A031927, A098974, A098976, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x + 1]]], {x, 1, 10000}]; a

q=2; forprime(p=3,1e5, if(p-q==24, print1(p", ")); q=p) \\ Charles R Greathouse IV, Mar 13 2020

a(n) = A098974(n) + 24. - Amiram Eldar, Mar 13 2020

a(n) >> n log^2 n. - Charles R Greathouse IV, Mar 13 2020

A271232 Composite integers sandwiched between primes p, q with q-p = 6.

Original entry on oeis.org

24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 74, 75, 76, 77, 78, 84, 85, 86, 87, 88, 132, 133, 134, 135, 136, 152, 153, 154, 155, 156, 158, 159, 160, 161, 162, 168, 169, 170, 171, 172, 174, 175, 176, 177, 178

Offset: 1

Michel Marcus, Apr 02 2016

nonn

			The composite number 24 is sandwiched between consecutive primes 23 and 29, and 29-23=6, so 24 is a member of the sequence.

Laurent Coppey, Décompositions multiplicatives directes des entiers, Diagrammes, 65-66 (2011), p. 1-68, in French, see J6 p. 11.

Cf. A014574, A031924, A031925, A045881, A271211.

lista(nn) = {forcomposite(c=4, nn, if ((p=precprime(c)) && ((nextprime(c)-p)==6), print1(c, ", ")););}

A052229 a(n) is the first prime p from A031924 such that A052180(primepi(p)) = prime(n).

Original entry on oeis.org

23, 47, 251, 167, 727, 433, 941, 1187, 1453, 1367, 2417, 4597, 2207, 3761, 4657, 4451, 5557, 6317, 7517, 8923, 9043, 17707, 15227, 12823, 10607, 33487, 28663, 29717, 50417, 31567, 24793, 24043, 28753, 28837, 29983, 29173, 59951, 45497

Offset: 3

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052158, A052180.

A052231 Primes p from A031924 such that A052180(primepi(p)) = 7.

Original entry on oeis.org

47, 73, 131, 157, 257, 367, 677, 971, 1097, 1123, 1181, 1543, 1601, 1753, 2383, 2441, 2467, 2677, 3307, 3407, 3617, 3727, 3911, 4357, 4457, 4903, 4987, 5113, 5297, 5381, 5407, 5743, 5801, 6037, 6373, 6977, 7187, 7213, 7481, 7717, 7817, 7901, 7927, 8053

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052180.

A052232 Primes p from A031924 such that A052180(primepi(p)) = 11.

Original entry on oeis.org

251, 647, 733, 977, 1063, 1657, 1901, 1987, 2713, 2957, 3637, 4211, 4871, 4937, 5683, 5861, 6257, 6673, 7247, 7577, 8831, 9491, 9643, 11801, 11953, 12197, 12613, 13121, 13451, 14923, 15101, 15187, 15761, 15913, 16421, 16487, 18223, 18797

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052180.

A052233 Primes p from A031924 such that A052180(primepi(p)) = 13.

Original entry on oeis.org

167, 373, 557, 607, 947, 1777, 2351, 2897, 4507, 5081, 5443, 5471, 6067, 7237, 8747, 9343, 9967, 10903, 11087, 12491, 12697, 13037, 14051, 15767, 15817, 16001, 16363, 16547, 16937, 16987, 17327, 19483, 21277, 24971, 26687, 26921, 30197, 30637

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052180.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A333200 Rectangular array read by antidiagonals: row n shows the primes p(k) such that p(k) = p(k-1) + 2n, with 2 prefixed to row 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A052159 Lower prime of a difference of 6 (G-minor-6 primes) between consecutive primes of 6k+5 form.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A052230 Primes p from A031924 such that A052180(primepi(p)) = 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A126720 Primes p such that p - q = 24, where q is the previous prime before p; or prime numbers preceded by precisely 23 composite numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A271232 Composite integers sandwiched between primes p, q with q-p = 6.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A052229 a(n) is the first prime p from A031924 such that A052180(primepi(p)) = prime(n).

Views

Author

Keywords

Crossrefs

A052231 Primes p from A031924 such that A052180(primepi(p)) = 7.

Views

Author

Keywords

Crossrefs

A052232 Primes p from A031924 such that A052180(primepi(p)) = 11.

Views

Author

Keywords

Crossrefs

A052233 Primes p from A031924 such that A052180(primepi(p)) = 13.

Views

Author

Keywords

Crossrefs