A098090 -id:A098090 - cp's OEIS frontend

A085090 If 2n-1 is prime then a(n) = 2n-1, otherwise a(n) = 0.

0, 3, 5, 7, 0, 11, 13, 0, 17, 19, 0, 23, 0, 0, 29, 31, 0, 0, 37, 0, 41, 43, 0, 47, 0, 0, 53, 0, 0, 59, 61, 0, 0, 67, 0, 71, 73, 0, 0, 79, 0, 83, 0, 0, 89, 0, 0, 0, 97, 0, 101, 103, 0, 107, 109, 0, 113, 0, 0, 0, 0, 0, 0, 127, 0, 131, 0, 0, 137, 139, 0, 0, 0, 0, 149, 151, 0, 0, 157, 0, 0, 163

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 02 2003

Previous name was: Starting with n+(n-1) go on adding n-2, then n-3, etc. until one gets a prime; a(n) = smallest prime in n+(n-1)+(n-2)+...+(n-i) (with the least i that gives a prime), or 0 if no such prime exists.

			a(8) = 0 as there is no prime in the partial sum of the finite sequence 8,7,6,5,4,3,2,1.
a(7) = 13 = 7 + 6.

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

Cf. A122845.

DoubleFactorial:=func< n | &*[n..2 by -2] >; [ DoubleFactorial(-5 + 4*n) mod (-1 + 2*n)^2: n in [1..90]]; // Vincenzo Librandi, Oct 04 2018

[IsPrime(2*n-1) select 2*n-1 else 0: n in [1..90]]; // Bruno Berselli, Oct 05 2018

apr[n_]:=Module[{cl=Select[Rest[Accumulate[Range[n,1,-1]]],PrimeQ, 1]}, If[cl=={},0,First[cl]]]; Array[apr,100] (* Harvey P. Dale, Jun 01 2012 *)
b[n_] := Mod[(-5 + 4 n)!!, (-1 + 2 n)^2]; a = Array[b, 82] (* Fred Daniel Kline, Oct 04 2018; Thomas Ordowski's formula with adjusted index *)

a(n) = if (isprime(p=2*n-1), p, 0); \\ Michel Marcus, Aug 09 2018

If 2n-1 is prime then a(n) = 2n-1, otherwise a(n) = 0. - David Wasserman, Jan 25 2005
a(A098090(n)-1)=2*A098090(n)-3; a(n)=(2*n-1)*A101264(n-1). - Reinhard Zumkeller, Sep 14 2006
a(n+1) = (4n-1)!! mod (2n+1)^2; by Gauss generalization of the Wilson's theorem. - Thomas Ordowski, Jul 23 2016

More terms from David Wasserman, Jan 25 2005

New name using formula from David Wasserman, Joerg Arndt, Jul 24 2016

A288313 Let b(k) denote A056240(k); the sequence lists numbers b(2n) where for all m > n, b(2m) > b(2*n).

Original entry on oeis.org

2, 4, 8, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753, 771, 789, 807, 813, 831, 843

Offset: 1

David James Sycamore, Jun 07 2017

nonn

This is an ascending subsequence of A056240 with even argument terms.
After the first three (even) terms, a(1) = b(2) = 2, a(2) = b(4) = 4, a(3) = b(6) = 8 respectively, all subsequent terms are odd (semiprime) numbers of the form 3*r, for r = primes 5, 7, 11, 13, .... The graph of all odd-valued terms a(n) for n >= 4 is a straight line (y = 3*x - 9), corresponding to b(2*n) = 3*(2*n) - 9 = 3*(2*n - 3) = 3*r, where r = 2*n - 3 is prime, and n is in sequence A098090. The sequence a(n) for n >= 4 is identical term for term to A001748(n) for n >= 3. In other words, for n >= 4, a(n) = 3*A000040(n-1).
If, for any even number n >= 6, n - 3 is prime, then A056240(n) belongs to this sequence.

			a(1) = 2 is included because for all n > 1, b(2n) > 2; likewise a(2) = b(4) = 4, and a(3) = b(6) = 8 are included. The first odd term, a(4) = b(8) = 15, is included since for all n > 4, b(2n) > 15. b(12) = 35 is not in this sequence because b(14) = 33 < 35, and only ascending terms are permitted.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A001748, A056240, A098090. Essentially the same as A063534.

Join[{2, 4, 8}, 3*Prime[Range[3, 100]]] (* Paolo Xausa, Apr 16 2024 *)

a(1) = 2, a(2) = 4, a(3) = 8, and for n >= 4, a(n) = 3*A000040(n-1).

Offset changed to 1 and entry edited to reflect this change by Michel Marcus, Jul 03 2017

A065305 Triangular array giving means of two odd primes: T(n,k) = (n-th prime + k-th prime)/2, n >= k >= 2.

Original entry on oeis.org

3, 4, 5, 5, 6, 7, 7, 8, 9, 11, 8, 9, 10, 12, 13, 10, 11, 12, 14, 15, 17, 11, 12, 13, 15, 16, 18, 19, 13, 14, 15, 17, 18, 20, 21, 23, 16, 17, 18, 20, 21, 23, 24, 26, 29, 17, 18, 19, 21, 22, 24, 25, 27, 30, 31, 20, 21, 22, 24, 25, 27, 28, 30, 33, 34, 37, 22, 23, 24, 26, 27, 29, 30

Offset: 2

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Oct 29 2001

			3; 4,5; 5,6,7; 7,8,9,11; ...

Reinhard Zumkeller, First 125 rows of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A065306.
a(n, k) = A065342(n, k)/2 [but note different offsets]
Cf. A098090 (left edge), A065091 (right edge), A000040.

import Data.List (inits)
a065305 n k = a065305_tabl !! (n-2) !! (k - 1)
a065305_row n = a065305_tabl !! (n-2)
a065305_tabl = zipWith (map . (flip div 2 .) . (+))
                       a065091_list $ tail $ inits a065091_list
-- Reinhard Zumkeller, Aug 02 2015, Jan 30 2012

seq(seq((ithprime(i)+ithprime(j))/2,j=2..i),i=2..20)

A153043 Numbers n > 1 such that 2*n-3 is not a prime.

Original entry on oeis.org

2, 6, 9, 12, 14, 15, 18, 19, 21, 24, 26, 27, 29, 30, 33, 34, 36, 39, 40, 42, 44, 45, 47, 48, 49, 51, 54, 57, 59, 60, 61, 62, 63, 64, 66, 68, 69, 72, 73, 74, 75, 78, 79, 81, 82, 84, 86, 87, 89, 90, 93, 94, 95, 96, 99, 102, 103, 104, 105, 106, 108, 109, 110

Offset: 1

Vincenzo Librandi, Dec 17 2008

One more than the associated value in A104275. - R. J. Mathar, Jan 05 2011
2*A155705(m,n)-3 = (2n+1)*(2m+1) are nonprime: all A155705(.,.) are in this sequence.
The terms after a(1) are the values of 2*h*k + k + h + 2, where h and k are positive integers. - Vincenzo Librandi, Jan 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A098090, A104275, A153040, A155705.

[n: n in [2..102] | not IsPrime(2*n-3)];  // Bruno Berselli, Mar 05 2011

Select[Range[2,200], !PrimeQ[2*#-3]&] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2012 *)

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

A182138 Irregular triangle T, read by rows, in which row n lists the distances between n and the two primes whose sum makes 2n in decreasing order (Goldbach conjecture).

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 4, 0, 5, 3, 4, 2, 7, 3, 8, 6, 0, 7, 5, 1, 10, 6, 0, 9, 3, 8, 4, 2, 13, 3, 14, 12, 6, 0, 13, 11, 5, 1, 12, 0, 17, 9, 3, 16, 10, 8, 2, 19, 15, 9, 20, 18, 6, 0, 19, 17, 13, 7, 5, 22, 18, 12, 6, 21, 15, 3, 20, 16, 14, 10, 4, 25, 15, 9, 24, 18, 12, 0, 23, 17, 13, 11, 7, 1

Offset: 2

Jean COHEN, Apr 16 2012

The Goldbach conjecture is that for any even integer 2n>=4, at least one pair of primes p and q exist such that p+q=2n. The present numbers listed here are the distances d between each prime and n, the half of the even integer 2n: d=n-p=q-n with p <= q.
See the link section for plots I added. - Jason Kimberley, Oct 04 2012
Each nonzero entry d of row n is coprime to n. For otherwise n+d would be composite. - Jason Kimberley, Oct 10 2012

			n=2, 2n=4, 4=2+2, p=q=2 -> d=0.
n=18, 2n=36, four prime pairs have a sum of 36: 5+31, 7+29, 13+23, 17+19, with the four distances d being 13=18-5=31-18, 11=18-7=29-18, 5=18-13=23-18, 1=18-17=19-18.
Triangle begins:
  0;
  0;
  1;
  2, 0;
  1;
  4, 0;
  5, 3;
  4, 2;
  7, 3;
  8, 6, 0;

Alois P. Heinz, Rows n = 2..600, flattened
OEIS (Plot 2), Plot of (n, d)
Subplots for fixed p:
OEIS (Plot 2), A067076 vs A098090 (p=3).
OEIS (Plot 2), A089038 vs A089253 (p=5).
OEIS (Plot 2), A105760 vs A089192 (p=7).
...
OEIS (Plot 2), A153143 vs A097932 (p=19).
Wikipedia, Goldbach's conjecture

Cf. A045917 (row lengths), A047949 (first column), A047160 (last elements of rows).

Cf. A184995.

A182138:= func;
&cat[A182138(n):n in [2..30]]; // Jason Kimberley, Oct 01 2012

T:= n-> seq(`if`(isprime(p) and isprime(2*n-p), n-p, NULL), p=2..n):
seq(T(n), n=2..40); # Alois P. Heinz, Apr 16 2012

T[n_] := Table[If[PrimeQ[p] && PrimeQ[2n-p], n-p, Nothing], {p, 2, n}];
Table[T[n], {n, 2, 30}] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

for(n=2,18,forprime(p=2,n,if(isprime(2*n-p),print1(n-p", ")))) \\ Charles R Greathouse IV, Apr 16 2012

T(n,i) = n - A184995(n,i). - Jason Kimberley, Sep 25 2012

A073273 a(n) = floor(sqrt(prime(n)*prime(n+2))).

Original entry on oeis.org

3, 4, 7, 9, 13, 15, 19, 23, 26, 32, 35, 39, 43, 47, 52, 56, 62, 65, 69, 74, 77, 83, 89, 94, 99, 103, 105, 109, 117, 121, 131, 134, 142, 144, 152, 156, 161, 167, 172, 176, 184, 186, 193, 195, 203, 210, 218, 225, 229, 233, 236, 244, 248, 256, 262, 266, 272, 275

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

A000040(n) < a(n) < A000040(n+2).

			prime(10)*prime(12) = 29*37 = 1073 = 32*32+49, therefore a(10)=32; A073274(10) = prime(11)-a(10) = 31-32 = -1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A073271, A073274, A078444.

[Floor(Sqrt(NthPrime(n)*NthPrime(n+2))): n in [1..60]]; // Vincenzo Librandi, Dec 12 2015

Table[Floor[Sqrt[Prime[n] Prime[n + 2]]], {n, 60}] (* Vincenzo Librandi, Dec 12 2015 *)

a(n) = sqrtint(prime(n)*prime(n+2)); \\ Michel Marcus, Dec 12 2015

a(n) = A098090(A028310(n - 1)) + A089038(n). - Miko Labalan, Dec 12 2015

A255707 Least number k > 0 such that (2*n-1)^k - 2 is prime, or 0 if no such number exists.

Original entry on oeis.org

0, 2, 1, 1, 1, 4, 1, 1, 6, 1, 1, 24, 1, 2, 2, 1, 1, 2, 2, 1, 4, 1, 1, 2, 1, 8, 4, 1, 12, 4, 1, 1, 8, 3, 1, 2, 1, 1, 2, 38, 1, 4, 1, 4, 2, 1, 2, 4, 747, 1, 4, 1, 1, 2, 1, 1, 10, 1, 2, 2, 2, 6, 42, 2, 1, 2, 1, 2, 10, 1, 1, 4, 2, 16, 50, 1, 1, 2, 22, 1, 2, 38

Offset: 1

Robert Price, Mar 02 2015

nonn

Michel Marcus, Table of n, a(n) for n = 1..152 (terms 1..143 from Robert Price)
Carlos Rivera, Puzzle 887. p(n)^c-2 is prime, The Prime Puzzles and Problems Connection.

Cf. A079706, A084712, A084713, A084714, A098090, A138066.

lst = {0}; For[n = 2, n ≤ 143, n++, For[k = 1, k >= 1, k++, If[PrimeQ[(2*n - 1)^k - 2], AppendTo[lst, k]; Break[]]]]; lst

a(n)=if(n==1,return(0));k=1;while(k,if(ispseudoprime((2*n-1)^k-2),return(k));k++)
vector(50,n,a(n)) \\ Derek Orr, Mar 03 2015

a(A098090(n)) = 1. - Michel Marcus, Mar 03 2015

A122845 Triangle read by rows, 3<=k<=n: T(n,k) = smallest prime p such that 2k-p and 2n-p are prime, T(n,k) = 0 if no such p exists.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 0, 5, 5, 5, 3, 3, 3, 7, 3, 3, 3, 3, 5, 3, 3, 0, 5, 5, 5, 7, 5, 5, 3, 3, 3, 7, 3, 3, 7, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 0, 5, 5, 5, 7, 5, 5, 7, 5, 5, 3, 3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 0, 5, 5, 5, 11, 5, 5, 17, 5, 5, 23, 5, 0, 0, 7, 7, 7, 11, 7, 7, 11, 7, 7, 11, 7, 3, 3, 3, 0, 3, 3, 13, 3, 3, 13

Offset: 3

Reinhard Zumkeller, Sep 14 2006

Eric Weisstein's World of Mathematics, Goldbach Partition
Index entries for sequences related to Goldbach conjecture

Cf. A098090.

T[n_, k_] := Module[{p}, For[p = 2, p < 2n && p < 2k, p = NextPrime[p], If[PrimeQ[2n - p] && PrimeQ[2k - p], Return[p]]]; 0];
Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Sep 22 2021 *)

T(A098090(n),3) = 2*A098090(n) - A085090(A098090(n)-1) = 3.

A143836 Triangle read by rows: T(r,c) = (prime(r+2) + prime(c+1))/2.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 8, 9, 10, 12, 10, 11, 12, 14, 15, 11, 12, 13, 15, 16, 18, 13, 14, 15, 17, 18, 20, 21, 16, 17, 18, 20, 21, 23, 24, 26, 17, 18, 19, 21, 22, 24, 25, 27, 30, 20, 21, 22, 24, 25, 27, 28, 30, 33, 34, 22, 23, 24, 26, 27, 29, 30, 32, 35, 36, 39, 23, 24, 25, 27, 28, 30, 31, 33, 36, 37, 40, 42

Offset: 1

Pierre CAMI, Sep 02 2008

The number of appearances of m >= 1 in this sequence is A061357(m). Conjecture: Every integer >= 4 appears at least once in this sequence. - Ya-Ping Lu, Mar 05 2023

The number of composites between 3 and (r+2)-th prime missing from Row 1 through Row r in the triangle is A334810(r+2). - Ya-Ping Lu, Mar 24 2023

			Triangle begins:
   4;
   5,  6;
   7,  8,  9;
   8,  9, 10, 12;
  10, 11, 12, 14, 15;
  ...

Cf. A098090 (1st column, except 1st term), A024675 (right diagonal).

T(r,c) = (prime(r+2) + prime(c+1))/2; \\ Michel Marcus, Mar 07 2023

Name simplified by Ya-Ping Lu, Mar 05 2023

cp's OEIS Frontend

A085090 If 2n-1 is prime then a(n) = 2n-1, otherwise a(n) = 0.

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A288313 Let b(k) denote A056240(k); the sequence lists numbers b(2*n) where for all m > n, b(2*m) > b(2*n).

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A065305 Triangular array giving means of two odd primes: T(n,k) = (n-th prime + k-th prime)/2, n >= k >= 2.

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A153043 Numbers n > 1 such that 2*n-3 is not a prime.

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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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Haskell

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A182138 Irregular triangle T, read by rows, in which row n lists the distances between n and the two primes whose sum makes 2n in decreasing order (Goldbach conjecture).

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Magma

Maple

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A073273 a(n) = floor(sqrt(prime(n)*prime(n+2))).

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A288313 Let b(k) denote A056240(k); the sequence lists numbers b(2n) where for all m > n, b(2m) > b(2*n).

A122845 Triangle read by rows, 3<=k<=n: T(n,k) = smallest prime p such that 2k-p and 2n-p are prime, T(n,k) = 0 if no such p exists.