A020483 -id:A020483 - cp's OEIS frontend

A101045 Record size primes in A101044.

2, 3, 5, 7, 13, 19, 31, 43, 53, 67, 79, 101, 149, 157, 163, 181, 197, 227, 307, 349, 379, 409, 431, 619, 631, 661, 691, 751, 757, 811, 829, 1093, 1117, 1217, 1279, 1423, 1453, 1481, 1531, 1549, 1579, 1759, 1877, 2239, 2273, 2287, 2383, 2447, 2659, 2671, 2707

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

This sequence (except 2) is also the record size primes in the longer A020483.

Conjecture: lim_{n->infinity} a(n)/n^2 = 1. - Ya-Ping Lu, Sep 24 2020

J. K. Andersen, Prime gaps (not necessarily consecutive).
Taras Goy and Mark Shattuck, Determinant Formulas of Some Hessenberg Matrices with Jacobsthal Entries, Applications and Appl. Math.: An Int'l J. (2021) Vol. 16, Issue 1, Art. 10.
Mike Oakes, Ed Pegg Jr, and Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101042, A101043, A101044, A101046.

from sympy import isprime, nextprime
m = p_max = 0
while m >= 0:
    p = 2
    while isprime(p + 2*m) == 0:
        p = nextprime(p)
    if p > p_max:
        print(p)
        p_max = p
    m += 1 # Ya-Ping Lu, Sep 24 2020

A101043 A101042 sorted. There exists a prime p for which a(n) is the smallest positive d such that p is the smallest prime where p+d is also prime.

Original entry on oeis.org

1, 2, 6, 22, 88, 112, 116, 202, 242, 284, 470, 718, 772, 1326, 1328, 1334, 1642, 1732, 1762, 2402, 2558, 3274, 5246, 5888, 7094, 7702, 7984, 9512, 9952, 9974, 10342, 10532, 12688, 13528, 16766, 25678, 25708, 37666, 59894, 60458, 61756, 62156

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

Except for n=1, A020483(a(n)/2) is the first appearance of a prime in A020483.

			d=6 is in the sequence because there exists the prime p=5 satisfying the required conditions: 2+6, 3+6 is composite and 5+6 is prime. 6 is the smallest such number.

J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101042, A101044, A101045, A101046.

A101044 Primes corresponding to A101043 (which is A101042 sorted).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 11, 31, 29, 23, 17, 43, 37, 41, 53, 47, 67, 79, 61, 71, 59, 73, 101, 149, 83, 127, 97, 89, 109, 137, 157, 107, 103, 151, 113, 163, 139, 181, 197, 131, 193, 167, 191, 173, 227, 199, 179, 223, 211, 307, 241, 349, 229, 239, 233, 257, 379, 277, 271

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

The order in which the primes (except 2) first appear in A020483. It is conjectured that all primes are in this sequence.

J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101042, A101043, A101045, A101046.

A101046 d such that the smallest prime p for which p+d is also prime is larger than for any smaller d.

Original entry on oeis.org

1, 2, 6, 22, 88, 112, 202, 718, 1328, 1642, 1732, 5246, 5888, 10342, 25678, 37666, 59894, 76004, 103102, 108412, 180814, 359662, 651362, 872698, 2373478, 6088792, 7642528, 9244552, 13038352, 13591192, 24318988, 34857778, 55076404, 147838742

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

The numbers in A101042 which are smaller than all following numbers.

			Consider d=6. The smallest prime p for which p+6 is also prime, is p=5. All numbers below d=6 have a p<5 (or no p at all), so 6 is in the sequence.

J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101042, A101043, A101044, A101045.

A086505 a(n) is the n-th smallest prime p such that p+2n is also prime.

Original entry on oeis.org

3, 7, 11, 23, 31, 29, 53, 73, 53, 89, 157, 73, 137, 199, 73, 281, 229, 127, 383, 229, 149, 389, 463, 193, 359, 547, 239, 467, 823, 197, 857, 883, 283, 809, 499, 389, 1013, 907, 421, 827, 1201, 373, 1151, 1231, 367, 1307, 1279, 577, 1229, 1009, 631, 1427, 1783

Offset: 1

Amarnath Murthy, Jul 29 2003

nonn

Note the patterns in the graph. These patterns depend on the number of prime factors of n: see color graph for different n's: n primes - black dots, n multiples of 3 - red dots, n multiples of 15 - green dots, n multiples of 105 - blue dots. - Zak Seidov, Nov 28 2013

Zak Seidov, Table of n, a(n) for n = 1..10000
Zak Seidov, Color graph for different n's

Cf. A020483, A231608.

N:= 10^4: # to get all terms before the first with a(n)+2*n > N
Primes:= select(isprime, {seq(2*i+1, i=1..N)}):
for n from 1 do
R:= Primes intersect map(`+`, Primes, -2*n);
if nops(R) < n then break fi;
A[n]:= R[n];
od:
seq(A[j],j=1..n-1); # Robert Israel, Aug 07 2014

Edited by Sam Alexander, Feb 26 2004

A108184 a(n) = smallest prime such that a(n) + 2n is also prime and such that a(n) > a(n-1).

Original entry on oeis.org

2, 3, 7, 11, 23, 31, 41, 47, 67, 71, 83, 109, 113, 131, 139, 149, 167, 193, 197, 233, 241, 251, 263, 271, 283, 317, 331, 347, 353, 373, 379, 401, 439, 443, 479, 487, 491, 503, 523, 541, 563, 571, 577, 587, 613, 619, 641, 727, 733, 761, 787, 809, 863, 877

Offset: 0

Giovanni Teofilatto, Jun 28 2005

Increasing primes p such that p + 2n is prime.

			a(0)=2 since 2+0=2 is prime; a(1)=3 since 3+2=5 is prime.
a(2)=7 since 7+4=11 is prime; 5 is not in the sequence since 5+4=9 is not prime.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A020483, A237055

A108184 := proc(n) option remember; if n = 1 then 3; else for a from procname(n-1)+1 do if isprime(a) and isprime(a+2*n) then RETURN(a) ; fi; od: fi; end: seq(A108184(n),n=1..100) ; # R. J. Mathar, Jan 31 2009

t = {2}; Do[p = NextPrime[t[[-1]]]; While[! PrimeQ[p + 2 n], p = NextPrime[p]]; AppendTo[t, p], {n, 100}]; t (* T. D. Noe, Feb 04 2014 *)

A108184(maxp) = {my(a=[2], n=1); forprime(p=3, maxp, if(isprime(p+2*n), n++; a=concat(a, p))); a} \\ Colin Barker, Feb 03 2014

Edited and extended by Ray Chandler, Jul 07 2005

Edited by N. J. A. Sloane, Feb 11 2009 at the suggestion of R. J. Mathar

A231608 Table whose n-th row consists of primes p such that p + 2n is also prime, read by antidiagonals.

Original entry on oeis.org

3, 3, 5, 5, 7, 11, 3, 7, 13, 17, 3, 5, 11, 19, 29, 5, 7, 11, 13, 37, 41, 3, 7, 13, 23, 17, 43, 59, 3, 5, 11, 19, 29, 23, 67, 71, 5, 7, 17, 17, 31, 53, 31, 79, 101, 3, 11, 13, 23, 19, 37, 59, 37, 97, 107, 7, 11, 13, 31, 29, 29, 43, 71, 41, 103, 137

Offset: 1

T. D. Noe, Nov 26 2013

			The following sequences are read by antidiagonals
{3, 5, 11, 17, 29, 41, 59, 71, 101, 107,...}
{3, 7, 13, 19, 37, 43, 67, 79, 97, 103,...}
{5, 7, 11, 13, 17, 23, 31, 37, 41, 47,...}
{3, 5, 11, 23, 29, 53, 59, 71, 89, 101,...}
{3, 7, 13, 19, 31, 37, 43, 61, 73, 79,...}
{5, 7, 11, 17, 19, 29, 31, 41, 47, 59,...}
{3, 5, 17, 23, 29, 47, 53, 59, 83, 89,...}
{3, 7, 13, 31, 37, 43, 67, 73, 97, 151,...}
{5, 11, 13, 19, 23, 29, 41, 43, 53, 61,...}
{3, 11, 17, 23, 41, 47, 53, 59, 83, 89,...}
...

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

Cf. A001359, A023200, A023201, A023202, A023203.
Cf. A046133, A153417, A049488, A153418, A153419.
Cf. A020483 (numbers in first column).
Cf. A086505 (numbers on the diagonal).

A231608 := proc(n,k)
    local j,p ;
    j := 0 ;
    p := 2;
    while j < k do
        if isprime(p+2*n ) then
            j := j+1 ;
        end if;
        if j = k then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
for n from 1 to 10 do
    for k from 1 to 10 do
        printf("%3d ",A231608(n,k)) ;
    end do;
    printf("\n") ;
end do: # R. J. Mathar, Nov 19 2014

nn = 10; t = Table[Select[Range[100*nn], PrimeQ[#] && PrimeQ[# + 2*n] &, nn], {n, nn}]; Table[t[[n-j+1, j]], {n, nn}, {j, n}]

A063713 Numbers n such that there exist primes p, q, r with n*2 = p - r = r + q (values of r are given in A063714).

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 27, 28, 30, 32, 33, 35, 36, 38, 39, 42, 43, 45, 46, 48, 50, 51, 52, 53, 54, 55, 57, 58, 60, 63, 65, 66, 67, 69, 70, 71, 72, 75, 77, 78, 80, 81, 84, 85, 87, 88, 90, 93, 96, 97, 98, 99, 100, 101, 102, 105

Offset: 1

Reinhard Zumkeller, Aug 10 2001

nonn

			10*2 = 20 = 23 - 3 = 3 + 17, A063714(7) = 3; 11*2 = 22 = 41 - 19 = 19 + 3, A063714(8) = 19 28 is missing because we have the prime sums (Goldbach): 5 + 23 = 11 + 17 and differences with primes less 28: 31 - 3 = 41 - 13 = 47 - 19; none of these have a prime in common.

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

Cf. A063714, A002373, A020483.

filter:= proc(n) local k;
  k:= 1;
  while k < 2*n do
    k:= nextprime(k);
    if isprime(2*n+k) and isprime(2*n-k) then return true fi
  od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Oct 09 2017

okQ[n_] := AnyTrue[Prime[Range[PrimePi[2 n - 2]]], PrimeQ[2 n + #] && PrimeQ[2 n - #]&]; Select[Range[105], okQ] (* Jean-François Alcover, Feb 12 2018 *)

A232558 a(1)=5, q=a(n) is the smallest prime > a(n-1) such that q-2*n = p prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 31, 47, 59, 61, 83, 97, 109, 131, 137, 139, 173, 193, 211, 233, 239, 241, 257, 271, 277, 281, 283, 307, 389, 397, 409, 431, 433, 457, 467, 491, 523, 563, 569, 571, 653, 661, 673, 701, 709, 733, 821, 823, 859, 887, 911, 967, 983, 991

Offset: 1

Pierre CAMI, Nov 26 2013

nonn

Conjecture: the sequence is infinite.

Remarks: the primes p appears in increasing order but with repetition, all the primes are not present in p,q.

			5-3=2 a(1)=5
7-3=4 a(2)=7
11-5=6 a(3)=11
13-5=8 a(4)=13
17-7=10 a(5)=17

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A020483

a[1]=5;a[n_]:=a[n]=(For[k=a[n-1]+2,!(k>2n&&PrimeQ[k]&&PrimeQ[k-2n]),k++];k)

A239392 Numbers n that have record value of prime p such that p + 2n is another prime.

Original entry on oeis.org

1, 3, 11, 44, 56, 101, 359, 664, 821, 866, 2623, 2944, 5171, 12839, 18833, 29947, 38002, 51551, 54206, 90407, 179831, 325681, 436349, 1186739, 3044396, 3821264, 4622276, 6519176, 6795596, 12159494, 17428889, 27538202, 73919371, 127586456, 266466008, 423717053, 458430559

Offset: 1

T. D. Noe, Mar 19 2014

nonn

See A101045 of the values of p > 2.

Cf. A020483 (least p with p, q both prime, such that p+2n = q).

Cf. A370998, A371069.

nn = 10^5; t = Table[j = 1; found = False; While[! found, j++; found = PrimeQ[Prime[j] + 2 i]]; Prime[j], {i, nn}]; mx = -1; t2 = {}; Do[If[t[[i]] > mx, mx = t[[i]]; AppendTo[t2, {i, t[[i]]}]], {i, nn}]; Transpose[t2][[1]]

a(n) = A370998(A371069(n)). - Hugo Pfoertner, Mar 11 2024

a(30)-a(35) from Giovanni Resta, Mar 19 2014

a(36)-a(37) from Hugo Pfoertner, Mar 11 2024

cp's OEIS Frontend

A101045 Record size primes in A101044.

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A101043 A101042 sorted. There exists a prime p for which a(n) is the smallest positive d such that p is the smallest prime where p+d is also prime.

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A101044 Primes corresponding to A101043 (which is A101042 sorted).

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A101046 d such that the smallest prime p for which p+d is also prime is larger than for any smaller d.

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A086505 a(n) is the n-th smallest prime p such that p+2n is also prime.

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A108184 a(n) = smallest prime such that a(n) + 2n is also prime and such that a(n) > a(n-1).

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A231608 Table whose n-th row consists of primes p such that p + 2n is also prime, read by antidiagonals.

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A063713 Numbers n such that there exist primes p, q, r with n*2 = p - r = r + q (values of r are given in A063714).

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A232558 a(1)=5, q=a(n) is the smallest prime > a(n-1) such that q-2*n = p prime.

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