A002092 -id:A002092 - cp's OEIS frontend

A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

0, 0, 1, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 15, 0, 5, 12, 3, 8, 9, 0, 7, 12, 3, 4, 15, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 15, 4, 3, 6, 5, 0, 9, 2, 15, 0, 5, 12, 3, 14, 9, 0, 7, 12, 9, 4, 15, 6, 7, 0, 9, 2, 3

Offset: 2

Lior Manor

I have confirmed there are no -1 entries through integers to 4.29*10^9 using PARI. - Bill McEachen, Jul 07 2008
From Daniel Forgues, Jul 02 2009: (Start)
Goldbach's Conjecture: for all n >= 2, there are primes (distinct or not) p and q s.t. p+q = 2n. The primes p and q must be equidistant (distance m >= 0) from n: p = n-m and q = n+m, hence p+q = (n-m)+(n+m) = 2n.
Equivalent to Goldbach's Conjecture: for all n >= 2, there are primes p and q equidistant (distance >= 0) from n, where p and q are n when n is prime.
If this conjecture is true, then a(n) will never be set to -1.
Twin Primes Conjecture: there is an infinity of twin primes.
If this conjecture is true, then a(n) will be 1 infinitely often (for which each twin primes pair is (n-1, n+1)).
Since there is an infinity of primes, a(n) = 0 infinitely often (for which n is prime).
(End)
If n is composite, then n and a(n) are coprime, because otherwise n + a(n) would be composite. - Jason Kimberley, Sep 03 2011
From Jianglin Luo, Sep 22 2023: (Start)
a(n) < primepi(n)+sigma(n,0);
a(n) < primepi(primepi(n)+n);
a(n) < primepi(n), for n>344;
a(n) = o(primepi(n)), as n->+oo. (End)
If -1 < a(n) < n-3, then a(n) is divisible by 3 if and only if n is not divisible by 3, and odd if and only if n is even. - Robert Israel, Oct 05 2023

			16-3=13 and 16+3=19 are primes, so a(16)=3.

T. D. Noe, Table of n, a(n) for n = 2..10000
Jason Kimberley, Symmetrical plot of A047160

Cf. A001031, A002092, A002372, A002373, A002374, A002375, A014092, A025583, A035026, A047949, A071406, A082467, A102084, A103147, A112823, A155764, A155765, A177461, A078611, A010051, A045917, A325142.

a047160 n = if null ms then -1 else head ms
            where ms = [m | m <- [0 .. n - 1],
                            a010051' (n - m) == 1, a010051' (n + m) == 1]
-- Reinhard Zumkeller, Aug 10 2014

A047160:=func;[A047160(n):n in[2..100]]; // Jason Kimberley, Sep 02 2011

Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, -1, k], {n, 2, 100}]
smm[n_]:=Module[{m=0},While[AnyTrue[n+{m,-m},CompositeQ],m++];m]; Array[smm,100,2] (* Harvey P. Dale, Nov 16 2024 *)

a(n)=forprime(p=n,2*n, if(isprime(2*n-p), return(p-n))); -1 \\ Charles R Greathouse IV, Jun 23 2017

10 N=2// 20 M=0// 30 if and{prmdiv(N-M)=N-M,prmdiv(N+M)=N+M} then print M;:goto 50// 40 inc M:goto 30// 50 inc N: if N>130 then stop// 60 goto 20

a(n) = n - A112823(n).

a(n) = A082467(n) * A005171(n), for n > 3. - Jason Kimberley, Jun 25 2012

More terms from Patrick De Geest, May 15 1999
Deleted a comment. - T. D. Noe, Jan 22 2009
Comment corrected and definition edited by Daniel Forgues, Jul 08 2009

A002091 From a Goldbach conjecture: the location of records in A185091.

Original entry on oeis.org

3, 9, 19, 21, 55, 115, 193, 323, 611, 1081, 1571, 10771, 13067, 16321, 44881, 57887, 93167, 189947, 404939, 442307, 1746551, 3383593, 3544391, 5056787, 7480667, 25619213, 87170987, 404940757, 526805663, 707095391, 1009465507, 1048720723, 5315914139

Offset: 1

N. J. A. Sloane

nonn

A stronger version of the second Goldbach conjecture (every odd number can be expressed as the sum of 3 primes) states that every odd number k > 5 can be written as k = 2*p + q, p, q prime. The conjecture was posed by E. Lemoine and later by H. Levy. The article by B. H. Mayoh assumes q {1,prime}. For the representations of k minimizing q, the sequence gives the value of k at which a larger q than for all representations of j < k is required. The new record value of q is given in A002092. The corresponding sequences for q prime and q=1 excluded are A194828 and A194829. - Hugo Pfoertner, Sep 03 2011
k is in this list when (k+1)/2 is the index of a record in A185091.
Checked up to k=10^13. a(50) is > 10^13. - Hugo Pfoertner, Sep 25 2011

			a(3)=19, because it is the first number for which q=5 is required. 3=2*1+1, 5=2*2+1, 7=2*3+1, 9=2*3+3, 11=2*5+1, 13=2*5+3, 15=2*7+1, 17=2*7+3, 19=2*7+5.

Brian H. Mayoh, On the second Goldbach conjecture, Nordisk Tidskr. Informations-Behandling 6, 1966, 48-50.
Emile Lemoine, L'intermédiaire des mathématiciens, 1 (1894), 179; ibid 3 (1896), 151.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..49
Brian H. Mayoh, On the second Goldbach conjecture, BIT Numerical Mathematics 6 (1966) 1, 48-50
Wikipedia, Lemoine's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A002092 [values of q], A194828, A194829 [equivalent with q=1 excluded].

Cf. A185091.

a(19)-a(32) from Hugo Pfoertner, Sep 03 2011
a(33) from Jason Kimberley, a(34)-a(40) from Hugo Pfoertner, Sep 09 2011
a(41)-a(49) from Hugo Pfoertner, Sep 25 2011

A194828 Odd numbers n>5 in the representation n=2*p+q, p, q prime, q minimal, at which a larger q than for any smaller n is needed. A194829 gives values of q.

Original entry on oeis.org

7, 11, 21, 23, 55, 83, 167, 611, 887, 1487, 1571, 10771, 12227, 13523, 16321, 44881, 54863, 57887, 93167, 189947, 404939, 442307, 1746551, 3383593, 3544391, 5056787, 6811307, 25619213, 87170987, 404940757, 526805663, 707095391, 1009465507, 1048720723

Offset: 1

Hugo Pfoertner, Sep 03 2011

nonn

Related to Lemoine's conjecture, similar to A002091, but with q=1 excluded. Checked up to 2*10^13.

			a(4)=23, because 23 can only be represented by 23=2*5+13, whereas in A002091 23=2*11+1 avoids the need of increasing q for this representation.

See A002091.

Hugo Pfoertner, Table of n, a(n) for n = 1..50
See A002091.

Cf. A002091, A002092, A194829, A195353.

A194829 Records of primes q in the representation of odd n>5 by n=2*p+q, p, q prime, q minimal. A194828 gives the values of n at which an increase of q is required.

Original entry on oeis.org

3, 5, 7, 13, 17, 37, 61, 73, 109, 181, 277, 317, 349, 397, 419, 503, 577, 601, 709, 829, 877, 1129, 1237, 1367, 1429, 1669, 1993, 2467, 2833, 2879, 3001, 3037, 3329, 3821, 4861, 5003, 5281, 5821, 5897, 6301, 6329, 6421, 7129, 7309, 7873, 8017, 8597, 8821, 8969, 9157

Offset: 1

Hugo Pfoertner, Sep 03 2011

nonn

See A002091. Checked up to n=2*10^13.

			a(5)=17, because it is the smallest possible value of q in the representation of 55=2*p+q. 55-3=52, 55-5=50, 55-7=48, 55-11=44, 55-13=42, none of which has the form 2*p. 55-17=38=2*19. All odd numbers < 55 can be represented using a q<17.

See A002091.

See A002091.

Cf. A194828, A002091, A002092 [q=1 allowed], A195354.

a(35)-a(43) from Hugo Pfoertner, Sep 11 2011
a(44)-a(49) from Hugo Pfoertner, Sep 18 2011
a(50) from Hugo Pfoertner, Sep 22 2011

A185091 The smallest positive noncomposite q such that 2n-1 = 2p+q for some positive noncomposite p.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 17, 11, 1, 3, 1, 3, 5, 7, 13, 11, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 17, 11, 1, 3, 5, 7, 29, 11, 1, 3, 1, 3, 5, 7, 13, 11, 1, 3, 5, 7, 1, 3, 1, 3, 5, 7, 13, 11, 1, 3, 5, 7, 1, 3, 5, 7, 17, 11, 1, 3, 5, 7, 29

Offset: 2

Jason Kimberley (with thanks to Hugo Pfoertner), Sep 05 2011

It is a Goldbach conjecture variant that terms exist for 2n-1 >= 5.
Lemma: N=2n-1 is coprime to q=a(n) unless N=3q. Proof: Suppose N and q are not coprime; so we have N=2p+q=iq with i=/=1=/=q, so (i-1)q=2p; now since q=/=2 (because N is odd), then q=p and i=3. QED.
Empirically, N=3q only for N=9,21.

Emile Lemoine, L'intermédiaire des mathématiciens, 1 (1894), 179; ibid 3 (1896), 151.

Jason Kimberley, Table of n, a(n) for n = 2..10002 (corrected by Michel Marcus, Jan 19 2019)
Brian H. Mayoh, On the second Goldbach conjecture, BIT Numerical Mathematics 6 (1966) 1, 48-50
Wikipedia, Lemoine's conjecture
Index entries for sequences related to Goldbach conjecture

Records in this sequence are in A002092 occurring at 2n-1 in A002091.

A185091 := func;[A185091(n):n in [3..94]];

cp's OEIS Frontend

A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

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A002091 From a Goldbach conjecture: the location of records in A185091.

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A194828 Odd numbers n>5 in the representation n=2*p+q, p, q prime, q minimal, at which a larger q than for any smaller n is needed. A194829 gives values of q.

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A194829 Records of primes q in the representation of odd n>5 by n=2*p+q, p, q prime, q minimal. A194828 gives the values of n at which an increase of q is required.

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A185091 The smallest positive noncomposite q such that 2n-1 = 2p+q for some positive noncomposite p.

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Magma