A031928 -id:A031928 - cp's OEIS frontend

A086138 Number of primes between p and p+10 if both p and (p+10) are prime, i.e., number of primes somewhere between 10+A023203(n) and A023203(n).

3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 0, 2, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 2

Offset: 1

Labos Elemer, Jul 29 2003

nonn

			a(n)=0,1,2,3 correspond to {p,p+10} prime-pairs either consecutive ones or those with various d-patterns like as follows: a(n)=0 to cases like 139[10]149; a(n)=2 to 7[4,2,4]17 etc.; a(n)=3 to one case 3[2,2,4,2]13 and a(n)=2 to cases like 31[6,4]37 or 43[4,6]53.

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

Cf. A023303, A031928, A031929.

cp[x_,y_] := Count[Table[PrimeQ[i],{i,x,y}],True] Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d],k=k+1; Print[cp[s+1,s+d-1]]],{n,1,1000}]; k; d=10
Count[Range[#+1,#+9],?PrimeQ]&/@Select[Prime[Range[500]],PrimeQ[#+10]&] (* _Harvey P. Dale, Jan 17 2025 *)

forprime(p=2,1e5,if(isprime(p+10), print1(isprime(p+2)+isprime(p+4)+isprime(p+6)+isprime(p+8)", "))) \\ Charles R Greathouse IV, May 15 2013

Definition clarified by Harvey P. Dale, Jan 17 2025

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

A231609 Table whose n-th row consists of primes p such that p + 2n is the next prime, read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Nov 26 2013

The plot has an unusual gap near 10^5. Why?

			The following sequences are read by antidiagonals
{   3,    5,   11,   17,   29,   41,   59,   71,  101,  107, ...}
{   7,   13,   19,   37,   43,   67,   79,   97,  103,  109, ...}
{  23,   31,   47,   53,   61,   73,   83,  131,  151,  157, ...}
{  89,  359,  389,  401,  449,  479,  491,  683,  701,  719, ...}
{ 139,  181,  241,  283,  337,  409,  421,  547,  577,  631, ...}
{ 199,  211,  467,  509,  619,  661,  797,  997, 1201, 1237, ...}
{ 113,  293,  317,  773,  839,  863,  953, 1409, 1583, 1847, ...}
{1831, 1933, 2113, 2221, 2251, 2593, 2803, 3121, 3373, 3391, ...}
{ 523, 1069, 1259, 1381, 1759, 1913, 2161, 2503, 2861, 3803, ...}
{ 887, 1637, 3089, 3413, 3947, 5717, 5903, 5987, 6803, 7649, ...}
...

T. D. Noe, Rows n = 1..100 of triangle, flattened

Cf. A001359, A029710, A031924, A031926, A031928.
Cf. A031930, A031932, A031934, A031936, A031938.
Cf. A000230 (numbers in first column).

nn = 10; t = Table[{}, {nn}]; complete = 0; lastP = 3; While[complete < nn, p = NextPrime[lastP]; diff = p - lastP; If[diff <= 2*nn && Length[t[[diff/2]]] < nn - diff/2 + 1, AppendTo[t[[diff/2]], lastP]; If[Length[t[[diff/2]]] == nn - diff/2 + 1, complete++]]; lastP = p]; t2 = PadRight[t, {nn, nn}, 0]; Table[t2[[n-j+1, j]], {n, nn}, {j, n}]

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A086138 Number of primes between p and p+10 if both p and (p+10) are prime, i.e., number of primes somewhere between 10+A023203(n) and A023203(n).

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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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A231609 Table whose n-th row consists of primes p such that p + 2n is the next prime, read by antidiagonals.

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A052257 Last filtering prime (A052180) of primes p such that next prime is p+10.

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