A020482 -id:A020482 - cp's OEIS frontend

A171637 Triangle read by rows in which row n lists the distinct primes of the distinct decompositions of 2n into unordered sums of two primes.

2, 3, 3, 5, 3, 5, 7, 5, 7, 3, 7, 11, 3, 5, 11, 13, 5, 7, 11, 13, 3, 7, 13, 17, 3, 5, 11, 17, 19, 5, 7, 11, 13, 17, 19, 3, 7, 13, 19, 23, 5, 11, 17, 23, 7, 11, 13, 17, 19, 23, 3, 13, 19, 29, 3, 5, 11, 17, 23, 29, 31, 5, 7, 13, 17, 19, 23, 29, 31, 7, 19, 31, 3, 11, 17, 23, 29, 37, 5, 11

Offset: 2

Juri-Stepan Gerasimov, Dec 13 2009

Each entry of the n-th row is a prime p from the n-th row of A002260 such that 2n-p is also prime. If A002260 is read as the antidiagonals of a square array, this sequence can be read as an irregular square array (see example below). The n-th row has length A035026(n). This sequence is the nonzero subsequence of A154725. - Jason Kimberley, Jul 08 2012

			a(2)=2 because for row 2: 2*2=2+2; a(3)=3 because for row 3: 2*3=3+3; a(4)=3 and a(5)=5 because for row 4: 2*4=3+5; a(6)=3, a(7)=5 and a(8)=7 because for row 5: 2*5=3+7=5+5.
The table starts:
2;
3;
3,5;
3,5,7;
5,7;
3,7,11;
3,5,11,13;
5,7,11,13;
3,7,13,17;
3,5,11,17,19;
5,7,11,13,17,19;
3,7,13,19,23;
5,11,17,23;
7,11,13,17,19,23;
3,13,19,29;
3,5,11,17,23,29,31;
As an irregular square array [_Jason Kimberley_, Jul 08 2012]:
3 . 3 . 3 . . . 3 . 3 . . . 3 . 3
. . . . . . . . . . . . . . . .
5 . 5 . 5 . . . 5 . 5 . . . 5
. . . . . . . . . . . . . .
7 . 7 . 7 . . . 7 . 7 . .
. . . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . .
11. 11. 11. . . 11
. . . . . . . .
13. 13. 13. .
. . . . . .
. . . . .
. . . .
17. 17
. .
19

Related triangles: A154720, A154721, A154722, A154723, A154724, A154725, A154726, A154727, A184995. - Jason Kimberley, Sep 03 2011

Cf. A020481 (left edge), A020482 (right edge), A238778 (row sums), A238711 (row products), A000040, A010051.

a171637 n k = a171637_tabf !! (n-2) !! (k-1)
a171637_tabf = map a171637_row [2..]
a171637_row n = reverse $ filter ((== 1) . a010051) $
   map (2 * n -) $ takeWhile (<= 2 * n) a000040_list
-- Reinhard Zumkeller, Mar 03 2014

Table[ps = Prime[Range[PrimePi[2*n]]]; Select[ps, MemberQ[ps, 2*n - #] &], {n, 2, 50}] (* T. D. Noe, Jan 27 2012 *)

Keyword:tabl replaced by tabf, arbitrarily defined a(1) removed and entries checked by R. J. Mathar, May 22 2010

Definition clarified by N. J. A. Sloane, May 23 2010

A060266 Difference between 2n and the following prime.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 5, 3, 1, 1, 5, 3, 1, 3, 1, 1, 3, 1, 5, 3, 1, 5, 3, 1, 1, 5, 3, 1, 3, 1, 1, 5, 3, 1, 3, 1, 5, 3, 1, 7, 5, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 13, 11, 9, 7, 5, 3, 1, 3, 1, 5, 3, 1, 1, 9, 7, 5, 3, 1, 1, 5, 3, 1, 5, 3, 1, 3, 1, 5, 3, 1, 5, 3, 1, 1, 9, 7, 5, 3, 1, 1, 3, 1, 1, 11, 9, 7, 5

Offset: 1

Labos Elemer, Mar 23 2001

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A020482, A049653, A060267, A060268, A060264.

with(numtheory): [seq(nextprime(2*i)-2*i,i=1..256)];

d2n[n_]:=Module[{c=2n},NextPrime[c]-c]; Array[d2n,120] (* Harvey P. Dale, May 14 2011 *)
Table[NextPrime@ # - # &[2 n], {n, 120}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = nextprime(2*n+1) - 2*n; \\ Michel Marcus, Feb 19 2017

Conjecture: Limit_{n->oo} (Sum_{k=1..n} a(k)) / (Sum_{k=1..n} log(2*k)) = 1. - Alain Rocchelli, Oct 24 2023

A118750 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 3*k. a(n) = product[k=1..n] A118749(k).

Original entry on oeis.org

3, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 2565568005, 74401472145, 2306445636495, 71499814731345, 2645493145059765, 108465218947450365, 4664004414740365695, 219208207492797187665, 10302785752161467820255

Offset: 1

Jonathan Vos Post, Apr 29 2006

Differs from (after first term) A048599 "Partial products of the sequence (A001097) of twin primes" after 8th term. Differs from (after first term) A070826 "One half of product of first n primes A000040" after 9th term. Analogous to A118455 a(1)=1. a(n) = product{k=1..n} P(k), where P(k) is the largest prime <= k.

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748.

A047949 a(n) is the largest m such that n-m and n+m are both primes, or -1 if no such m exists.

Original entry on oeis.org

0, 0, 1, 2, 1, 4, 5, 4, 7, 8, 7, 10, 9, 8, 13, 14, 13, 12, 17, 16, 19, 20, 19, 22, 21, 20, 25, 24, 23, 28, 29, 28, 27, 32, 31, 34, 35, 34, 33, 38, 37, 40, 39, 38, 43, 42, 41, 30, 47, 46, 49, 50, 49, 52, 53, 52, 55, 54, 53, 48, 51, 50, 45, 62, 61, 64, 63, 62, 67, 68, 67, 66

Offset: 2

Lior Manor

A067076 is a subsequence of this sequence: when 2m+3 is prime a(m+3) = m. Moreover, it is the subsequence of records (maximal increasing subsequence): let m=a(n), with p=n-m and q=p+2m both odd primes > 3; now 3+2(m+(p-3)/2)=q and hence a(3+m+(p-3)/2) >= m+(p-3)/2 > m = a(n) but 3+m+(p-3)/2 < n. - Jason Kimberley, Aug 30 2012 and Oct 10 2012
Goldbach's conjecture says a(n) >= 0 for all n. - Robert Israel, Apr 15 2015
a(n) is the Goldbach partition of 2n which results in the maximum spread divided by 2. - Robert G. Wilson v, Jun 18 2018

			49-30=19 and 49+30=79 are primes, so a(49)=30.

T. D. Noe, Table of n, a(n) for n = 2..10000
OEIS (Plot 2), A067076 vs A098090 (n-m=3). - _Jason Kimberley_, Oct 01 2012

Cf. A047160, A067076, A182138.
Cf. A020481.
Cf. A010051.

a047949 n = if null qs then -1 else head qs  where
   qs = [m | m <- [n, n-1 .. 0], a010051' (n+m) == 1, a010051' (n-m) == 1]
-- Reinhard Zumkeller, Nov 02 2015

a:= proc(n)
local k;
  for k from n - 1 to 0 by -2 do
     if isprime(n+k) and isprime(n-k) then return(k) fi
od:
-1
end proc:
0, seq(a(n),n=3..1000); # Robert Israel, Apr 16 2015

a[2] = a[3] = 0; a[n_] := (For[m = n - 2, m >= 0, m--, If[PrimeQ[n - m] && PrimeQ[n + m], Break[]]]; m); Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Sep 04 2013 *)
lm[n_]:=Module[{m=n-2},While[!AllTrue[n+{m,-m},PrimeQ],m--];m]; Join[{0,0}, Array[ lm,70,4]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 03 2014 *)
f[n_] := Block[{q = 2}, While[q <= n && !PrimeQ[2n -q], q = NextPrime@ q]; n - q]; Array[f, 72, 2] (* Robert G. Wilson v, Jun 18 2018 *)

a(n) = {if (n==2 || n==3, return (0)); my(m = 1, lastm = -1, do = 1); while (do, if (isprime(n-m) && isprime(n+m), lastm = m); m++; if (m == n - 1, do = 0);); return (lastm);} \\ Michel Marcus, Jun 09 2013

a(n)=if(n<4,0,forprime(p=3,n-1,if(isprime(2*n-p),return(n-p)));-1) \\ Ralf Stephan, Dec 29 2013

a(n) = n - A020481(n).

a(n) = (A020482(n) - A020481(n))/2. - Gionata Neri, Apr 15 2015

Corrected by Harvey P. Dale, Dec 21 2000

A060268 Distance of 2n from the closest prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 5, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 5, 3, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 5, 3, 1, 1, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1

Offset: 2

Labos Elemer, Mar 23 2001

nonn

			n=13, 2n=26 surrounded by 23 and 29 which are from 26 in equal distance of 3 and min{3,3}=3=a(13).

Cf. A020482, A049653, A051702, A059959, A060266.

with(numtheory): [seq(min(nextprime(2*i)-2*i, 2*i-prevprime(2*i)), i=2...256)];

a[n_] := Min[NextPrime[2*n] - 2*n, 2*n - NextPrime[2*n, -1]]; Array[a, 100, 2] (* Amiram Eldar, Sep 16 2020 *)

a(n) = min(2*n - precprime(2*n-1), nextprime(2*n+1) - 2*n); \\ Michel Marcus, Sep 16 2020

a(n) = min(A049653(n), A060266(n)). - Michel Marcus, Sep 16 2020

A238711 Product of all primes p such that 2n - p is also prime.

Original entry on oeis.org

2, 3, 15, 105, 35, 231, 2145, 5005, 4641, 53295, 1616615, 119301, 21505, 7436429, 21489, 57998985, 3038795305, 4123, 13844919, 10393190665, 12838371, 299859855, 7292509103495, 12023917269, 70691995, 37198413949697, 62483343, 2769282065, 98755025688454681

Offset: 2

Reinhard Zumkeller, Mar 06 2014

nonn

Product of n-th row in triangle A171637;
All terms greater than 3 are odd, composite and squarefree numbers, cf. A024556.
n is prime iff n is a factor of a(n).
Product of the distinct primes in the Goldbach partitions of 2n. - Wesley Ivan Hurt, Sep 29 2020

Reinhard Zumkeller, Table of n, a(n) for n = 2..2000

Cf. A000040, A010051, A238778, subsequence of A056911.

a238711 n = product $ filter ((== 1) . a010051') $
   map (2 * n -) $ takeWhile (<= 2 * n) a000040_list

Table[Times@@Select[Select[Prime[Range[2 n]], # < 2 n &], PrimeQ[2 n - #] &], {n, 2, 30}] (* Robert Price, Apr 26 2025 *)

A020639(a(n)) = A020481(n); A006530(a(n)) = A020482(n);
A001221(a(n)) = A035026(n); A008472(a(n)) = A238778(n);
A027748(a(n),k) + A027748(a(n),l+1-k) = 2*n for k=1..l, with l=A001221(a(n)); particulary A020639(a(n))+A006530(a(n)) = 2*n;
a(n) = n^c(n) * Product_{i=1..n-1} (i*(2*n-i))^(c(i)*c(2*n-i)), where c is the prime characteristic (A010051). - Wesley Ivan Hurt, Sep 29 2020

A060267 Difference between 2 closest primes surrounding 2n.

Original entry on oeis.org

2, 2, 4, 4, 2, 4, 4, 2, 4, 4, 6, 6, 6, 2, 6, 6, 6, 4, 4, 2, 4, 4, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 4, 4, 2, 6, 6, 6, 4, 4, 6, 6, 6, 8, 8, 8, 8, 4, 4, 2, 4, 4, 2, 4, 4, 14, 14, 14, 14, 14, 14, 14, 4, 4, 6, 6, 6, 2, 10, 10, 10, 10, 10, 2, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 2, 10, 10, 10, 10, 10, 2, 4

Offset: 2

Labos Elemer, Mar 23 2001

nonn

			a(3) = 2 because the closest primes to 2*3 = 6 are (5,7) and the difference between these is 2. - _Michael De Vlieger_, Nov 02 2017

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Cf. A020482, A049653, A060264, A060265, A060266, A060268.

with(numtheory): [seq(nextprime(2*i)-prevprime(2*i),i=2..256)];

Array[Subtract @@ NextPrime[#, {1, -1}] &[2 #] &, 96, 2] (* Michael De Vlieger, Nov 02 2017 *)
NextPrime[#]-NextPrime[#,-1]&/@(2*Range[2,100]) (* Harvey P. Dale, Nov 07 2017 *)

a(n) = nextprime(2*n+1) - precprime(2*n-1); \\ Michel Marcus, Sep 16 2020

A101778 Last term in each row of triangle referenced in A101777.

Original entry on oeis.org

3, 5, 3, 2, 7, 5, 3, 7, 7, 5, 3, 2, 11, 7, 7, 5, 3, 13, 11, 7, 7, 5, 3, 2, 13, 13, 11, 7, 7, 5, 3, 17, 13, 13, 11, 7, 7, 5, 3, 2, 19, 17, 13, 13, 11, 7, 7, 5, 3, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 2, 23, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 23, 23, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 2, 23

Offset: 1

Ray Chandler, Jan 10 2005

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A100555, A101776, A101777.

A020482(k) = forprime(q=2, k, if(isprime(2*k-q), return(2*k-q)));
a(n) = {my(r=(ceil(sqrt(2*n+1)))^2-2*n+3); if(r%2==0, r=A020482(r/2), if(isprime(r-2), r-=2, r=A020482(r\2))); r; } \\ Jinyuan Wang, Jan 29 2020

a(n) = A101777(A000217(n)).

A118747 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 2*k. a(n) = product[k=1..n] A060308(k).

Original entry on oeis.org

2, 6, 30, 210, 1470, 16170, 210210, 2732730, 46456410, 882671790, 16770764010, 385727572230, 8871734161290, 204049885709670, 5917446685580430, 183440847252993330, 5686666264842793230, 176286654210126590130

Offset: 1

Jonathan Vos Post, Apr 29 2006

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748.

A118752 a(n) = product[k=0..n] P(k), where P(k) is the smallest prime > 3*n. a(n) = product[k=0..n] A118751(k).

Original entry on oeis.org

2, 10, 70, 770, 10010, 170170, 3233230, 74364290, 2156564410, 62540367890, 1938751404590, 71733801969830, 2654150672883710, 108820177588232110, 4679267636293980730, 219925578905817094310, 11656055682008305998430

Offset: 0

Jonathan Vos Post, Apr 29 2006

Analogous to A118456 a(n) = product{k=1..n} P(k), where P(k) is the smallest prime >= k.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748-51.

Rest[FoldList[Times,1,Table[NextPrime[3n],{n,0,20}]]] (* Harvey P. Dale, Mar 09 2014 *)

Definition corrected by Harvey P. Dale, Mar 09 2014

cp's OEIS Frontend

A171637 Triangle read by rows in which row n lists the distinct primes of the distinct decompositions of 2n into unordered sums of two primes.

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A060266 Difference between 2n and the following prime.

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A118750 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 3*k. a(n) = product[k=1..n] A118749(k).

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A047949 a(n) is the largest m such that n-m and n+m are both primes, or -1 if no such m exists.

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A060268 Distance of 2n from the closest prime.

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A238711 Product of all primes p such that 2n - p is also prime.

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A060267 Difference between 2 closest primes surrounding 2n.

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A101778 Last term in each row of triangle referenced in A101777.