A031930 -id:A031930 - cp's OEIS frontend

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

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

A320704 Indices of primes followed by a gap (distance to next larger prime) of 12.

Original entry on oeis.org

46, 47, 91, 97, 114, 121, 139, 168, 197, 203, 214, 232, 239, 240, 242, 267, 278, 280, 290, 312, 317, 342, 357, 363, 376, 381, 404, 423, 437, 439, 449, 452, 461, 470, 472, 489, 499, 511, 546, 550, 562, 565, 599, 600, 617, 633, 634, 647, 653, 657, 675, 680, 692, 698, 716, 728

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031930.

Vincenzo Librandi, Table of n, a(n) for n = 1..10150
Index entries for primes, gaps between.

Equals A000720 o A031930.
Row 6 of A174349.
Indices of 12'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..1000] | NthPrime(n+1) - NthPrime(n) eq 12]; // Vincenzo Librandi, Mar 21 2019

Select[Range[1000], Prime[#] + 12 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 21 2019 *)

A320704_vec(N=100,g=12,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(A031930(n)).

A320704 = { i > 0 | prime(i+1) = prime(i) + 12 }.

A379239 Numbers k for which A003961(k)-sigma(k) is prime, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

4, 6, 7, 10, 12, 13, 15, 19, 21, 22, 23, 28, 31, 33, 34, 35, 37, 39, 43, 45, 47, 48, 51, 53, 55, 58, 61, 67, 73, 76, 77, 79, 82, 83, 84, 89, 95, 97, 103, 105, 109, 111, 112, 113, 115, 118, 123, 124, 127, 129, 131, 141, 142, 143, 145, 148, 151, 153, 155, 156, 157, 159, 161, 163, 165, 167, 173, 185, 187, 192, 193, 199

Offset: 1

Antti Karttunen, Dec 23 2024

nonn

			10 is included as A003961(10)-sigma(10) = 21-18 = 3 which is prime.
13 is included as A003961(13)-sigma(13) = 17-14 = 3 which is prime.
23 is included as A003961(23)-sigma(23) = 29-24 = 5 which is prime.

Cf. A000203, A003961, A286385, A379238 (characteristic function).
Subsequences: A023200, A031924, A031926, A031930, A031932, A031936, A031938, etc, i.e., all primes for which the gap to the next prime is one more than some prime.
Cf. also A349165.

PARI
```
is_A379239 = A379238;
```

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}]

A052258 Last filtering prime (A052180) of primes p such that next prime is p+12.

Original entry on oeis.org

11, 13, 11, 11, 17, 23, 17, 19, 17, 29, 13, 13, 19, 37, 29, 29, 11, 23, 31, 31, 11, 11, 29, 31, 29, 43, 11, 29, 43, 37, 19, 31, 13, 47, 17, 31, 43, 19, 31, 13, 61, 11, 53, 11, 47, 43, 37, 17, 19, 13, 71, 11, 41, 23, 61, 37, 19, 17, 41, 61, 19, 17, 59, 13, 47, 79, 37, 13, 71

Offset: 1

Labos Elemer, Feb 02 2000

nonn

Cf. A031930, A052180.

A368640 a(0) = 2; for n > 0, a(n) is the smallest prime that differs from the next prime by 2n and is not part of a run of 3 or more consecutive primes in arithmetic progression, or -1 if no such prime exists.

Original entry on oeis.org

2, 11, 7, 23, 89, 139, 467, 113, 1831, 523, 887, 1129, 1669, 2477, 2971, 4297, 5591, 1327, 9551, 30593, 19333, 16141, 15683, 81463, 28229, 31907, 19609, 35617, 82073, 44293, 43331, 34061, 89689, 162143, 134513, 173359, 31397, 404597, 212701, 188029, 542603, 265621

Offset: 0

Jon E. Schoenfield, Jan 01 2024

nonn

Equivalently, a(0) = 2, and for n > 0, a(n) is the smallest prime p such that NextPrime(p) - p = 2n but neither p - PreviousPrime(p) nor NextPrime(NextPrime(p)) - NextPrime(p) is equal to 2n, or -1 if no such prime p exists.
Stated yet another way: define an array A such that A(i,j) is the smallest prime that is at the start of a run of exactly i consecutive prime gaps of size exactly j, or -1 if no such prime exists; then a(0) = 2 and, for n > 0, a(n) = A(1,2n).
Conjecture: a(n) is never -1.

			a(1) is the smallest prime that differs from the next prime by 2 and is not part of a run of 3 or more consecutive primes in arithmetic progression. (Thus a(1) is the smallest prime p that is the lesser of a twin prime pair (p,q) (cf. A001359) where neither p nor q is part of another twin prime pair.) The primes 3, 5, 7 are in arithmetic progression with difference 2, so none of those primes is a(1); the smallest prime that meets the requirements is 11, so a(1) = 11.
a(6) is the smallest prime that differs from the next prime by 12 and is not part of a run of 3 or more consecutive primes in arithmetic progression. Primes that differ from the next prime by 12 are 199, 211, 467, 509, 619, ... (cf. A031930), but 199 and 211 are part of a run of three consecutive primes in arithmetic progression (199, 211, and 223). Thus, since PreviousPrime(467) != 455 and NextPrime(479) != 491, 467 is not in a run of three or more consecutive primes in arithmetic progression, so a(6) = 467.

Cf. A000040, A000230, A001359, A006560, A031930.

a(n) >= A000230(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A320704 Indices of primes followed by a gap (distance to next larger prime) of 12.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A379239 Numbers k for which A003961(k)-sigma(k) is prime, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A052258 Last filtering prime (A052180) of primes p such that next prime is p+12.

Views

Author

Keywords

Crossrefs

A368640 a(0) = 2; for n > 0, a(n) is the smallest prime that differs from the next prime by 2n and is not part of a run of 3 or more consecutive primes in arithmetic progression, or -1 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula