A031926 -id:A031926 - cp's OEIS frontend

A083371 Primes p such that q-p >= 8, where q is the next prime after p.

89, 113, 139, 181, 199, 211, 241, 283, 293, 317, 337, 359, 389, 401, 409, 421, 449, 467, 479, 491, 509, 523, 547, 577, 619, 631, 661, 683, 691, 701, 709, 719, 743, 761, 773, 787, 797, 811, 829, 839, 863, 887, 911, 919, 929, 953, 983, 997, 1021, 1039, 1051, 1069

Offset: 1

Benoit Cloitre, Jun 04 2003

nonn

The original definition by Cloitre was: [Start from any initial value F(1) >= 2 and define F(n) as the largest prime factor of F(1)+F(2)+F(3)+...+F(n-1). The sequence contains the primes satisfying F(2*p)=p supposed F(1)=7. Conjecture: F(n)= n/2+O(log n) and the sequence is infinite.] Don Reble showed Jan 22 2022 that these are the same primes p followed by a prime gap of q-p >=8, where q is the next prime after p: [
Let X' be the first prime after X, 'X be the first prime before X.
The F sequence starting at "7" has 11 "7"s, then 6 "11"s, 6 "13"s, 6 "17"s, 6 "19"s, 10 "23"s, ...
One easily sees that the F sequence starting at prime S has S' instances of S; then for each prime P after S, it has (P'-'P) instances of P.  (A076973 is the F sequence starting at "2".)
The primes from S to P occupy the first [S' + (S''-S) + (S'''-S') + ... + (P' - 'P)] terms of F.
That sum telescopes to P'+P-S, and so
    F(P'+P-S) = P;  F(P'+P-S+1) = P';
    F(P+'P-S) = 'P; F(P+'P-S+1) = P.
If F(X) =P, then P+'P-S < X   <= P'+P-S.
If F(2P)=P, then P+'P-S < 2P  <= P'+P-S
                     'P < P+S <= P'
                            S <= P'-P
So this sequence has the primes P for which P'-P >= 7; and since P'-P is even (both primes are odd), P'-P >= 8. q.e.d.]

Nathaniel Johnston, Table of n, a(n) for n = 1..1500
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A076973.

d:=8; M:=1000; t0:=[]; for n from 1 to M do p:=ithprime(n); if nextprime(p) - p >= d then t0:=[op(t0),p]; fi; od: t0; # N. J. A. Sloane, Dec 19 2006
f := proc(n) option remember: if(n=1)then return 7: fi: return max(op(numtheory[factorset](add(f(i),i=1..n-1)))): end: seq(`if`(f(2*ithprime(n))=ithprime(n),ithprime(n),NULL),n=1..200); # Nathaniel Johnston, Jun 25 2011, via Cloitre's F

Transpose[Select[Partition[Prime[Range[200]],2,1],Last[#]-First[#]>7&]][[1]] (* Harvey P. Dale, Jan 28 2013 *)

A000040 MINUS A124590. - R. J. Mathar, Jan 23 2022

A031926 UNION A031928 UNION A031930 UNION A031932 UNION ... - R. J. Mathar, Jan 23 2022

Terms after a(20) from Nathaniel Johnston, Jun 26 2011

Merged with A124583 in response to Reble's seqfan post. - R. J. Mathar, Jan 24 2022

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 *)

A049436 p, p+8 and either p+2 or p+6 or both are all primes.

Original entry on oeis.org

3, 5, 11, 23, 29, 53, 59, 71, 101, 131, 149, 173, 191, 233, 263, 269, 431, 563, 569, 593, 599, 653, 821, 1013, 1031, 1061, 1223, 1229, 1283, 1289, 1319, 1451, 1481, 1601, 1613, 1619, 1871, 2081, 2129, 2333, 2339, 2381, 2543, 2549, 2711, 2789, 2963, 3251

Offset: 1

Labos Elemer

nonn

			3 is here because 5, 7 and 11 are primes; 5 is here because 7, 11 and 13 are primes.

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

A031926, A023202, A049437, A049438, A046134, A046138, A007530.

Select[Prime[Range[500]],PrimeQ[#+8]&&AnyTrue[#+{2,6},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 04 2017 *)

A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 12, 15, 17, 22, 25, 29, 32, 39, 44, 45, 60, 65, 67, 72, 75, 80, 82, 94, 95, 99, 100, 109, 114, 117, 120, 124, 127, 137, 152, 155, 164, 169, 172, 177, 185, 194, 199, 204, 205, 214, 215, 220, 229, 240, 242, 247, 254, 260, 262, 267, 269, 270, 289, 304, 312, 330, 334, 347, 355, 359, 369, 374, 379, 389

Offset: 1

Sally Myers Moite, Apr 14 2019

nonn

There are 140 such numbers between 1 and 1000.
These numbers correspond to all the prime pairs which differ by 8 except 3 and 11.
Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd + c - d, 6cd - c + d or 6cd + c + d - 1, that is, they are not (6c - 1)d - c - 1, (6c - 1)d + c, (6c + 1)d - c or (6c + 1)d + c - 1.

			a(4) = 5, so 6(5) - 1 = 29 and 6(5) + 7 = 37 are both prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Sally M. Moite, “Primeless” Sieves for Primes and for Prime Pairs Which Differ by 2m, vixra:1903.0353 (2019).

The primes are A023202, A092402, A031926.
Similar sequences for twin primes are A002822, A067611, for "cousin" primes A056956, A186243.
Intersection of A024898 and A153218.
Cf. also A307561, A307562.

select(t -> isprime(6*t-1) and isprime(6*t+7), [$1..500]); # Robert Israel, May 27 2019

isok(n) = isprime(6*n-1) && isprime(6*n+7); \\ Michel Marcus, Apr 16 2019

A339920 Primes p such that p^2 - p*q + q^2 is prime, where q is the next prime after p.

Original entry on oeis.org

2, 3, 53, 151, 167, 263, 373, 443, 467, 509, 523, 571, 1063, 1103, 1117, 1217, 1493, 1553, 1901, 1973, 2161, 2207, 2281, 2399, 2713, 2837, 2963, 3259, 3347, 3511, 4073, 4297, 4373, 4463, 4523, 4673, 4691, 4877, 5147, 5237, 5303, 5419, 5471, 5851, 6211, 6311, 6367

Offset: 1

Michel Marcus, Dec 23 2020

nonn

From Bernard Schott, Dec 23 2020: (Start)
Except for a(2)=3, (3, 5) gives A339698(2) = 19, there is no other pair of twin primes (p, p+2) (p in A001359) that gives a prime number of the form p^2-p*q+q^2 = p^2+2p+4.
There are no consecutive cousin primes (p, p+4) (p in A029710) that gives a prime number of the form p^2-pq+q^2 = p^2+4p+16.
There are no consecutive primes with a gap of 8 (p, p+8) (p in A031926) that give a prime number of the form p^2-pq+q^2 = p^2+8p+64. (End)

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

Cf. A339698.

q:= 2: count:= 0: R:= NULL:
while count < 100 do
  p:= q; q:= nextprime(q);
  if isprime(p^2-p*q+q^2) then
    count:= count+1; R:= R, p;
  fi
od:
R; # Robert Israel, Dec 24 2020

forprime(p=1, 1e4, my(q=nextprime(p+1)); if(ispseudoprime(p^2-p*q+q^2), print1(p, ", ")));

A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

Original entry on oeis.org

389, 479, 1559, 3209, 8669, 12269, 12401, 13151, 14411, 14759, 21851, 28859, 31469, 33191, 36551, 39659, 40751, 50321, 54311, 64601, 70229, 77339, 79601, 87671, 99551, 102539, 110261, 114749, 114761, 118661, 129449, 132611, 136511

Offset: 1

Labos Elemer, Mar 22 2000

nonn

A subsequence of A031926. [Corrected by Sean A. Irvine, Nov 07 2021]
a(n)=p, the initial prime of two consecutive 8-twins of primes as follows: [p,p+8] and [p+12,p+12+8], d=8, while the distance of the two 8-twins is 12 (minimal; see A052380(4/2)=12).
Analogous sequences are A047948 for d=2, A052378 for d=4, A052376 for d=10 and A052188-A052199 for d=6k, so that in the [d,D-d,d] difference patterns which follows a(n) the D-d is minimal(=0,2,4; here it is 4).

			p=1559 begins the [1559,1567,1571,1579] prime quadruple consisting of two 8-twins [1559,1567] and[1571,1579] which are in minimal distance, min{D}=1571-1559=12=A052380(8/2).

Cf. A031926, A053325, A052380, A052376, A052378, A052188-A052190.

a(n) is the initial term of a [p, p+8, p+12, p+12+8] quadruple of consecutive primes.

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

A204813 Primes followed by a gap of 256 = nextprime(p)-p.

Original entry on oeis.org

1872851947, 2362150363, 2394261637, 2880755131, 2891509333, 3353981623, 3512569873, 3727051753, 3847458487, 4008610423, 4486630573, 4541745583, 4755895531, 4837532347, 5227869607, 5389475977, 6201260587, 6229685347, 6952228483, 7325665111, 7414468513

Offset: 1

M. F. Hasler, Jan 19 2012

nonn

Washington Bomfim, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between.

Cf. A204812, A204670, A126784, A031934, A031926, A029710, A001359; A062529, A138198, A123995.

list_gaps(g=256,f,N=25,p=0)=for(c=1,N,while(g+p!=p=nextprime(p+1),);if(f,write(f".txt",c" ",p-g),print1(", "p-g)))

a(8)-a(21) from Washington Bomfim

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;
```

A132253 Isolated primes congruent to 29 (mod 30).

Original entry on oeis.org

89, 359, 389, 449, 479, 509, 719, 839, 929, 1109, 1259, 1409, 1439, 1499, 1559, 1709, 1889, 1979, 2039, 2069, 2099, 2399, 2459, 2579, 2609, 2699, 2819, 2879, 2909, 2939, 3089, 3209, 3449, 3659, 3719, 3779, 3989, 4079, 4139, 4289, 4349

Offset: 1

Omar E. Pol, Aug 20 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A007510, A031926, A118922.

Select[Range[29,4500,30],PrimeQ[#]&&NoneTrue[#+{2,-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 13 2018 *)

cp's OEIS Frontend

A083371 Primes p such that q-p >= 8, where q is the next prime after p.

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

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A049436 p, p+8 and either p+2 or p+6 or both are all primes.

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Mathematica

A307563 Numbers k such that both 6k - 1 and 6k + 7 are prime.

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A339920 Primes p such that p^2 - p*q + q^2 is prime, where q is the next prime after p.

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A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

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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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A204813 Primes followed by a gap of 256 = nextprime(p)-p.

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