A101043 -id:A101043 - cp's OEIS frontend

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

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.

A020483 Least prime p such that p+2n is also prime.

Original entry on oeis.org

2, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 7, 5, 3, 5, 3, 7, 5, 3, 13, 7, 5, 3, 5, 3, 3, 5, 3, 3, 5, 3, 19, 13, 11, 13, 7, 5, 3, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 5, 3, 3, 13, 11, 31, 7

Offset: 0

David W. Wilson

nonn

It is conjectured that a(n) always exists. a(n) has been computed for n < 5 * 10^11, with largest value a(248281210271) = 3307. - Jens Kruse Andersen, Nov 28 2004

If a(n) = a(n+1) = k, then 2*n + k and 2*(n+1) + k are twin primes. - Ya-Ping Lu, Sep 22 2020

			Given n = 2, we see that 2 + 2n = 6 = 2 * 3, but 3 + 2n = 7, which is prime, so a(2) = 3.
Given n = 3, we see that 2 + 2n = 8 = 2^3 and 3 + 2n = 9 = 3^2, but 5 + 2n = 11, which is prime, so a(3) = 5.

T. D. Noe, Table of n, a(n) for n = 0..10000
Jens Kruse Andersen, Prime gaps (not necessarily consecutive), Yahoo! group "primenumbers", Nov 26 2004.
Jens Kruse Andersen, Mike Oakes, Ed Pegg Jr, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004.

Cf. A087711, A101042, A101043, A101044, A101045, A101046.
Cf. A101045, A239392 (record values).
Cf. A000040, A010051, A020484, A237055.
It is likely that A054906 is an identical sequence, although this seems to have not yet been proved. - N. J. A. Sloane, Feb 06 2017

P:=Filtered([1..10000],IsPrime);;
a:=List(List([0..110],n->Filtered(P,i->IsPrime(i+2*n))),Minimum); # Muniru A Asiru, Mar 26 2018

a020483 n = head [p | p <- a000040_list, a010051' (p + 2 * n) == 1]
-- Reinhard Zumkeller, Nov 29 2014

A020483 := proc(n)
    local p;
    p := 2;
    while true do
        if isprime(p+2*n) then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc:
seq(A020483(n),n=0..40); # R. J. Mathar, Sep 23 2016

Table[j = 1; found = False; While[!found, j++; found = PrimeQ[Prime[j] + 2i]]; Prime[j], {i, 200}]
leastPrimep2n[n_] := Block[{k = 1, p, q = 2 n}, While[p = Prime@k; !PrimeQ[p + q], k++]; p]; Array[leastPrimep2n, 102] (* Robert G. Wilson v, Mar 26 2008 *)

a(n)=forprime(p=2,,if(isprime(p+2*n), return(p))) \\ Charles R Greathouse IV, Mar 19 2014

If a(n) exists, a(n) < 2n, which of course is a great overestimate. - T. D. Noe, Jul 16 2002
a(n) = A087711(n) - n. - Zak Seidov, Nov 28 2007
a(n) = A020484(n) - 2n. - Zak Seidov, May 29 2014
a(n) = 2 if and only if n = 0. - Alonso del Arte, Mar 14 2018

a(0)=2 added by N. J. A. Sloane, Apr 25 2015

A101042 a(n) is the smallest positive d such that the n-th prime is the smallest prime p for which p+d is also prime.

Original entry on oeis.org

1, 2, 6, 22, 116, 88, 470, 112, 284, 242, 202, 772, 1326, 718, 1334, 1328, 2558, 1762, 1642, 2402, 3274, 1732, 7094, 9512, 7984, 5246, 12688, 10532, 9952, 16766, 7702, 60458, 9974, 25708, 5888, 13528, 10342, 25678, 62156, 69518, 76838, 37666

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

Except for n=1, A020483(a(n)/2) is the first appearance of the n-th prime. It is conjectured that a(n) always exists. a(386) is the first number which must be above 10^12.

			a(3)=6 because: The 3rd prime is 5. 2+6, 3+6 is composite, 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, A101043, A101044, A101045, A101046.

A101045 Record size primes in A101044.

Original entry on oeis.org

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

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.

cp's OEIS Frontend

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

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A020483 Least prime p such that p+2n is also prime.

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A101042 a(n) is the smallest positive d such that the n-th prime is the smallest prime p for which p+d is also prime.

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A101045 Record size primes in A101044.

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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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