A020484 -id:A020484 - cp's OEIS frontend

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

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

A073704 Smallest prime p such that also p-prime(n)*2 is a prime.

Original entry on oeis.org

7, 11, 13, 17, 29, 29, 37, 41, 53, 61, 67, 79, 89, 89, 97, 109, 131, 127, 137, 149, 149, 163, 173, 181, 197, 233, 211, 227, 223, 229, 257, 269, 277, 281, 311, 307, 317, 331, 337, 349, 389, 367, 389, 389, 397, 401, 433, 449, 457, 461, 479, 491, 487, 509, 521

Offset: 1

Reinhard Zumkeller, Aug 04 2002

nonn

Cf. A073703, A001747, A000040, A020484.

A127019 a(n) is the least semiprime s such that s - 2*n is also a semiprime.

Original entry on oeis.org

6, 10, 10, 14, 14, 21, 35, 22, 22, 26, 26, 33, 35, 34, 34, 38, 38, 46, 77, 46, 46, 58, 55, 57, 65, 58, 58, 62, 62, 69, 77, 74, 87, 74, 74, 82, 95, 82, 82, 86, 86, 93, 95, 94, 94, 106, 115, 106, 119, 106, 106, 118, 115, 118, 119, 118, 118, 122, 122, 129, 143, 133, 141, 134

Offset: 1

Rick L. Shepherd, Jan 02 2007

nonn

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

Cf. A127018, A020484.

N:= 500: # for terms before the first term > N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]):
SP:= select(`<=`,{seq(seq(P[i]*P[j],i=1..j),j=1..nops(P))},N):
R:= NULL:
for n from 1 do
  Q:= SP intersect (SP +~ (2*n));
  if Q = {} then break fi;
  R:= R, min(Q)
od:
R; # Robert Israel, Jan 30 2025

sp=Select[Range[1000],PrimeOmega[#]==2&];seq={};Do[s=0;Until[t=sp[[s]]-2n;PrimeOmega[t]==2&&t>0,s++];AppendTo[seq,sp[[s]]],{n,64}];seq (* James C. McMahon, Dec 30 2024 *)

A155882 Smallest positive prime number such that a(n)-2n is also prime, a(n) < a(n+1), and the differences a(n)-2n must increase with n.

Original entry on oeis.org

5, 11, 17, 31, 41, 53, 61, 83, 89, 103, 131, 137, 157, 167, 179, 199, 227, 233, 271, 281, 293, 307, 317, 331, 367, 383, 401, 409, 431, 439, 463, 503, 509, 547, 557, 563, 577, 599, 619, 643, 653, 661, 673, 701, 709, 733, 821, 829, 859, 887, 911, 967, 983, 991

Offset: 1

Eric Angelini, Jan 29 2009

Subtracting from a(1) twice n=1 gives 5-2=3, which is a prime number; subtracting from a(2) twice n=2 gives 11-4=7, which is a prime number; subtracting from a(3) twice n=3 gives 17-6=11, which is a prime number; subtracting from a(4) twice n=4 gives 31-8=23, which is a prime number; etc.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A020484, A108184 (for the differences a(n)-2n).

b:= proc(n) option remember; global a; a(n); b(n) end: a:= proc(n) option remember; local m; global b; if n=1 then b(1):= 3; 5 else for m from a(n-1)+2 by 2 while not (isprime(m) and (b(n-1)Alois P. Heinz, Feb 05 2009

Corrected definition and more terms from Alois P. Heinz, Feb 05 2009

A206770 Smallest number k such that sigma(k-2n)=sigma(k)-2n.

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 29, 29, 21, 31, 37, 37, 37, 41, 28, 33, 47, 47, 53, 53, 53, 59, 59, 44, 61, 67, 67, 67, 71, 57, 73, 79, 79, 79, 83, 83, 69, 89, 74, 101, 68, 97, 97, 85, 101, 103, 107, 107, 109, 113, 93, 131, 127, 127, 131, 127, 127, 127

Offset: 1

Paolo P. Lava, Jan 10 2013

nonn

Note all k>=1 are considered, even if k-2n<0. If the search space is k>=2n, variants of A020484 and A060264 appear. - R. J. Mathar, Jan 12 2013

			a(15)=37 because 37 is the minimum number for which sigma(37-2*15)=sigma(7)=8 and sigma(37)-2*15=38-30=8.

Cf. A054906

A206770:=proc(q)
local k,n;
for n from 1 to q do
for k from 1 to q do
  if sigma(k-2*n)=sigma(k)-2*n then print(k); break; fi;
od; od; end:
A206770(1000000000);
A206770 := proc(n)
    local k ;
    for k from 1 do
        if numtheory[sigma](k-2*n) = numtheory[sigma](k)-2*n then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 12 2013

A107257 Smallest prime p such that for each j <= n there are primes a < b <= p whose difference b - a is 2*j.

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 17, 19, 23, 23, 29, 29, 29, 31, 37, 37, 37, 41, 41, 43, 47, 47, 53, 53, 53, 59, 59, 59, 61, 67, 67, 67, 71, 71, 73, 79, 79, 79, 83, 83, 89, 89, 89, 101, 101, 101, 101, 101, 101, 103, 107, 107, 109, 113, 113, 131, 131, 131, 131, 131, 131, 131, 131

Offset: 1

Klaus Brockhaus, May 15 2005

nonn

Every positive even number <= 2*n is the difference of two suitable primes <= a(n).

Sequence is nondecreasing, whereas the related sequence A020484 is not; first divergence is at 45: a(45) = 101, A020484(45) = 97.

			Consider n = 45: 89, 97, 101 are consecutive primes, 2*45 = 97 - 7, but 2*44 = 101 - 13 cannot be written as b - a where a and b are primes <=97, hence a(45) = 101.

Cf. A060264, A020484.

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

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A073704 Smallest prime p such that also p-prime(n)*2 is a prime.

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A127019 a(n) is the least semiprime s such that s - 2*n is also a semiprime.

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Maple

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A155882 Smallest positive prime number such that a(n)-2n is also prime, a(n) < a(n+1), and the differences a(n)-2n must increase with n.

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Maple

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A206770 Smallest number k such that sigma(k-2*n)=sigma(k)-2*n.

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Comments

Examples

Crossrefs

Programs

Maple

A107257 Smallest prime p such that for each j <= n there are primes a < b <= p whose difference b - a is 2*j.

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Author

Keywords

Comments

Examples

Crossrefs

A206770 Smallest number k such that sigma(k-2n)=sigma(k)-2n.