A092953 -id:A092953 - cp's OEIS frontend

A040976 a(n) = prime(n) - 2.

0, 1, 3, 5, 9, 11, 15, 17, 21, 27, 29, 35, 39, 41, 45, 51, 57, 59, 65, 69, 71, 77, 81, 87, 95, 99, 101, 105, 107, 111, 125, 129, 135, 137, 147, 149, 155, 161, 165, 171, 177, 179, 189, 191, 195, 197, 209, 221, 225, 227, 231, 237, 239, 249, 255, 261

Offset: 1

Felice Russo

Numbers k such that k! reduced mod (k+2) is 1. - Benoit Cloitre, Mar 11 2002
The first a(n) numbers starting from 2 are divisible by primes up to prime(n-1). - Lekraj Beedassy, Jun 21 2006
The terms in this sequence are the cumulative sums of distances from one prime to another. For example for the distance from the first to 26th prime, 2 to 101, the cumulative sum of distances is 99, always the last prime, here 101, minus 2. - Enoch Haga, Apr 24 2006
The primes in this sequence are the initial primes of twin prime pairs. - Sebastiao Antonio da Silva, Dec 21 2008
Note that many, but not all, of these numbers satisfy x such that x^(x+1) = 1 mod (x+2). The first exception is 339. - Thomas Ordowski, Nov 27 2013
If this sequence had an infinite number of primes, the twin prime conjecture would follow. Sequence holds all primes in A001359. - John W. Nicholson, Apr 14 2014
From Bernard Schott, Feb 19 2023: (Start)
Equivalently, except for a(1)=0, all terms are odd integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.
For each term d, there exists only one such AP of primes, and this one always starts with A342309(d) = 2, so this unique AP is (2, 2+d) = (2, prime(m)) with m > 1; so, first examples are (2,3), (2,5), (2,7), (2,11), ... next elements should be respectively: 4, 8, 12, 20, ... that are all composite numbers.
Similar sequence with even common differences d is A360735.
This subsequence of A359408 corresponds to the first case: '2 is prime'; second case corresponding to the even common differences d is A360735. (End)

			a(13) = 39, because A000040(13) = 41.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
S. A. Khan, Primes in Geometric-Arithmetic Progression, arXiv preprint arXiv:1203.2083 [math.NT], 2012.
S. M. Ruiz and J. Sondow, Formulas for pi(n) and the n-th prime, arXiv:math/0210312 [math.NT], 2002-2014.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A000040, A001223, A014689, A014692.
Equals A359408 \ A360735.
Cf. A123556, A342309.
First column of A086800, and of A379011, last diagonal of A090321, and of A162621.
See also irregular triangles A103728, A319148, A369497.

Filtered([1..10^2],IsPrime)-2; # Muniru A Asiru, Jan 31 2018

a040976 n = a000040 n - 2
a040976_list = map (subtract 2) a000040_list
-- Reinhard Zumkeller, Feb 22 2012

[NthPrime(n)-2: n in [1..60]]; // Vincenzo Librandi, Jan 31 2018

seq(ithprime(n)-2,n=1..100); # Muniru A Asiru, Jan 31 2018

Prime[Range[22]]-2 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=prime(n)-2 \\ Charles R Greathouse IV, Nov 20 2012

a(n) = A000040(n) - 2 = Sum_{i=1..n-1} A001223(i).
For n > 2: A092953(a(n)) = 1. - Reinhard Zumkeller, Nov 10 2012
If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(16) = 2 and 16 is not a term. - Bernard Schott, Feb 19 2023
a(n) = Sum_{k = 2..floor(2n*log(n)+2)} (1-floor(A000720(k)/n)). [Ruiz and Sondow]. - Elias Alejandro Angulo Klein, Apr 09 2024

A007921 Numbers that are not the difference of two primes.

Original entry on oeis.org

7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 79, 83, 85, 89, 91, 93, 97, 103, 109, 113, 115, 117, 119, 121, 123, 127, 131, 133, 139, 141, 143, 145, 151, 153, 157, 159, 163, 167, 169, 173, 175, 181, 183, 185, 187, 193

Offset: 1

R. Muller

Conjecturally, odd numbers k such that k+2 is composite.
Is this the same as A068780(2n-1) - 1? - J. Stauduhar, Aug 23 2012
A092953(a(n)) = 0. - Reinhard Zumkeller, Nov 10 2012
It seems that the sequence contains the squares of all primes except for 2 and 3. - Ivan N. Ianakiev, Aug 29 2013 [It does: For every prime p > 3, note that p^2 == 1 (mod 3), so p^2 cannot be q - r where q and r are primes. (If it were, then since p^2 is odd, q and r could not both be odd primes; r would have to be the even prime, 2, which would mean that p^2 = q - 2, so q = p^2 + 2 == 0 (mod 3), i.e., 3 would divide q, so q would not be prime -- a contradiction.) - Jon E. Schoenfield, May 03 2024]
Integers d such that A123556(d) = 1, that is, integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has only one element. For each such d, the unique element of all the first largest APs with 1 element is A342309(d) = 2. - Bernard Schott, Jan 08 2023
If it exists, the least even term is > 10^12 (see 1st comment in A020483). - Bernard Schott, Jan 09 2023

F. Smarandache, Properties of Numbers, 1972. (See Smarandache odd sieve.)

T. D. Noe, Table of n, a(n) for n = 1..10000
Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
C. Dumitrescu and V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
F. Smarandache, Only Problems, Not Solutions!, 4th ed., 1993, Problem 94.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.
Index entries for primes, gaps between.

Cf. A048859.
Complement of A030173. Cf. A001223.
Cf. also A005408, A010051.
Cf. A123556, A342309.
Largest AP of prime numbers with k elements: this sequence (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), A206042 (k=8), A206043 (k=9), A206044 (k=10), A206045 (k=11).

a007921 n = a007921_list !! (n-1)
a007921_list = filter ((== 0) . a010051' . (+ 2)) [1, 3 ..]
-- Reinhard Zumkeller, Jul 03 2015

filter :=  d -> irem(d, 2) <> 0 and not isprime(2+d) : select(filter, [`$`(1 .. 200)]); # Bernard Schott, Jan 08 2023

Lim=200;nn=10;seq:=Complement[Range[Lim],Union[Flatten[Differences/@Subsets[Prime[Range[nn]],{2}]]]];Until[AllTrue[seq,OddQ],nn++];seq (* James C. McMahon, May 04 2024 *)

is(n)=n%2 && !isprime(n+2) \\ On Polignac's conjecture; Charles R Greathouse IV, Jun 28 2013

from sympy import isprime
print([n for n in range(1, 200) if n%2 and not isprime(n + 2)]) # Indranil Ghosh, Jun 15 2017, after Charles R Greathouse IV

A061357 Number of 0

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 4, 1, 3, 4, 3, 3, 5, 4, 3, 5, 3, 3, 6, 2, 5, 6, 2, 5, 6, 4, 5, 7, 4, 4, 8, 4, 4, 9, 4, 4, 7, 3, 6, 8, 5, 5, 8, 6, 7, 10, 6, 5, 12, 3, 5, 10, 3, 7, 9, 5, 5, 8, 7, 7, 11, 5, 5, 12, 4, 8, 11, 4, 8, 10, 5, 5, 13, 9, 6, 11, 7, 6, 14, 6, 8, 13, 5, 8, 11, 6, 9

Offset: 1

Amarnath Murthy, Apr 28 2001

Number of prime pairs (p,q) with p < n < q and q-n = n-p.
The same as the number of ways n can be expressed as the mean of two distinct primes.
Conjecture: for n>=4 a(n)>0. - Benoit Cloitre, Apr 29 2003
Conjectures from Rick L. Shepherd, Jun 24 2003: (Start)
1) For each integer N>=1 there exists a positive integer m(N) such that for n>=m(N) a(n)>a(N). (After the first m(N)-1 terms, a(N) does not reappear). In particular, for N=1 (or 2 or 3), m(N)=4 and a(N)=0, giving Benoit Cloitre's conjecture. (cont.)
(cont.) Conjectures based upon observing a(1),...,a(10000):
m(4)=m(5)=m(6)=m(7)=m(19)=20 for a(4)=a(5)=a(6)=a(7)=a(19)=1,
m(8)=...(7 others)...=m(34)=35 for a(8)=...(7 others)...=a(34)=2,
m(12)=...(10 others)...=m(64)=65 for a(12)=...(10 others)...=a(64)=3,
m(18)=...(10 others)...=m(79)=80 for a(18)=...(10 others)...=a(79)=4,
m(24)=...(14 others)...=m(94)=95 for a(24)=...(14 others)...=a(94)=5,
m(30)=...(17 others)...=m(199)=200 for a(30)=...(17 others)...=a(199)=6, etc.
2) Each nonnegative integer appears at least once in the current sequence.
3) Stronger than 2): A001477 (nonnegative integers) is a subsequence of the current sequence. (Supporting evidence: I've observed that 0,1,2,...,175 is a subsequence of a(1),...,a(10000)).
(End)
a(n) is also the number of k such that 2*k+1=p and 2*(n-k-1)+1=q are both odd primes with p < q with p*q = n^2 - m^2. [Pierre CAMI, Sep 01 2008]
Also: Number of ways n^2 can be written as b^2+pq where 0a(n) = sum (A010051(2*n - p): p prime < n). [Reinhard Zumkeller, Oct 19 2011]
a(n) is also the number of partitions of 2*n into two distinct primes. See the first formula by T. D. Noe, and the Alois P. Heinz, Nov 14 2012, crossreference. - Wolfdieter Lang, May 13 2016
All 0Jamie Morken, Jun 02 2017
a(n) is the number of appearances of n in A143836. - Ya-Ping Lu, Mar 05 2023

			a(10)= 2: there are two such pairs (3,17) and (7,13), as 10 = (3+17)/2 = (7+13)/2.

Pierre CAMI, Table of n, a(n) for n = 1..60000

Cf. A071681 (subsequence for prime n only).
Cf. A092953.
Bisection of A117929 (even part). - Alois P. Heinz, Nov 14 2012

a061357 n = sum $
   zipWith (\u v -> a010051 u * a010051 v) [n+1..] $ reverse [1..n-1]
-- Reinhard Zumkeller, Nov 10 2012, Oct 19 2011

Table[Count[Range[n - 1], k_ /; And[PrimeQ[n - k], PrimeQ[n + k]]], {n, 98}] (* Michael De Vlieger, May 14 2016 *)

a(n)=my(s);forprime(p=2,n-1,s+=isprime(2*n-p));s \\ Charles R Greathouse IV, Mar 08 2013

from sympy import primerange, isprime
def A061357(n): return sum(1 for p in primerange(n) if isprime((n<<1)-p)) # Chai Wah Wu, Sep 03 2024

a(n) = A045917(n) - A010051(n). - T. D. Noe, May 08 2007
a(n) = sum(A010051(n-k)*A010051(n+k): 1 <= k < n). - Reinhard Zumkeller, Nov 10 2012
a(n) = sum_{i=2..n-1} A010051(i)*A010051(2n-i). [Wesley Ivan Hurt, Aug 18 2013]

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

A030173 Differences p(i)-p(j) between primes, sorted in numerical order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 27, 28, 29, 30, 32, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 59, 60, 62, 64, 65, 66, 68, 69, 70, 71, 72, 74, 76, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90

Offset: 1

Alexander Grasser [Graesser] (alex(AT)computicket.com)

Conjectured (Polignac 1849) to be union of even numbers and the odd primes minus 2.

For n > 2: A092953(a(n)) > 0. - Reinhard Zumkeller, Nov 10 2012

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Complement of A007921. Cf. A001223, A005843, A040976.

import Data.List.Ordered (union)
a030173 n = a030173_list !! (n-1)
a030173_list = union [2, 4 ..] $ tail a040976_list
-- Reinhard Zumkeller, Jul 03 2015

nn = 90; Union[Range[2, nn, 2], Prime[Range[2, PrimePi[nn+2]]] - 2]

print1(1);p=3;forprime(q=5,1e3,forstep(n=p-1,q-3,2,print1(", "n));print1(", ",q-2);p=q) \\ conjectural; Charles R Greathouse IV, Jul 02 2011

isOK(n)=if(n%2,isprime(n+2),forprime(p=3,,isprime(n+p)&&return(1)));
for(n=1,10^100,isOK(n)&print1(n,", ")) \\ unconditionally outputs correct values only, will "hang" forever if conjecture is false once that exceptional even number is reached; Jeppe Stig Nielsen, Sep 23 2015

A202472 Goldbach's Problem extended to subtraction: number of decompositions of 2n into unordered differences of two primes, p, q, where p < 2n < q.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 3, 3, 2, 6, 4, 3, 6, 3, 4, 6, 4, 5, 8, 4, 4, 7, 6, 4, 9, 8, 4, 11, 5, 5, 11, 6, 8, 9, 4, 7, 11, 7, 4, 13, 7, 5, 15, 7, 8, 13, 8, 9, 11, 7, 7, 13, 10, 5, 13, 7, 7, 19, 9, 8, 17, 9, 10, 16, 9, 9, 15, 12, 7, 19, 9, 7, 19, 9, 12, 17, 8, 14

Offset: 1

James D. Klein, Dec 19 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Extension of A002375.

Bisection of A092953.

Table[Length[Select[Prime[Range[PrimePi[2*n]]], PrimeQ[2*n + #] &]], {n, 100}] (* T. D. Noe, Apr 16 2013 *)

a(n)=my(s);forprime(p=2,2*n,s+=isprime(2*n+p));s \\ Charles R Greathouse IV, Dec 19 2011
(C++)
#include 
using namespace std;
int main()
{ int p[25] = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97};
  int count, istart = 2;
  for(int n=1; n<=25; n++)
  {
      if(2*n>p[istart]) istart++;
      count = 0;
      for(int j=1; p[j]<2*n; j++)
        for(int i=istart; p[i]-p[j]<=2*n; i++)
          if(p[i]-p[j]==2*n) count++;
      cout << n << ". " << count << endl;
  }
    return 0;
} // code for the first 25 integers, James D. Klein, Dec 21 2011

a(n) = A092953(2*n). - Bill McEachen, May 24 2024

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cp's OEIS Frontend

A092954 Index of the first occurrence of n in A092953. There are a(n) primes p < n such that n+p is a prime.

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A040976 a(n) = prime(n) - 2.

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A007921 Numbers that are not the difference of two primes.

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A061357 Number of 0

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A030173 Differences p(i)-p(j) between primes, sorted in numerical order.

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A202472 Goldbach's Problem extended to subtraction: number of decompositions of 2n into unordered differences of two primes, p, q, where p < 2n < q.

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