A194828 -id:A194828 - cp's OEIS frontend

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.

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

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

A002092 From a Goldbach conjecture: records in A185091.

Original entry on oeis.org

1, 3, 5, 7, 17, 29, 47, 61, 73, 83, 277, 317, 349, 419, 503, 601, 709, 829, 877, 1129, 1237, 1367, 1429, 1669, 1801, 2467, 2833, 2879, 3001, 3037, 3329, 3821, 4861, 5003, 5281, 5821, 5897, 6301, 6329, 6421, 6481, 6841, 7069, 7121, 7309, 7873, 8017, 8597, 8821

Offset: 1

N. J. A. Sloane

nonn

See A002091. The sequence gives the record values of q in the representations minimizing q of 2*k+1 = 2*p+q, p prime, q {1,prime}.

Checked up to 2*k = 10^13.

Brian H. Mayoh, On the second Goldbach conjecture. Nordisk Tidskr. Informations-Behandling 6, 1966, pp. 48-50.
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).

See A002091.
Index entries for sequences related to Goldbach conjecture

Cf. A002372, A002373, A002374, A002375, A045917, A006307, A002091, A185091, A194828, A194829.

a(n) = A185091((A002091(n)+1)/2).

Comment added, 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

A195353 Odd numbers n>5 in the representation n=2*p+q, p, q prime, q maximal, at which a larger p than for any smaller n is needed. A195354 gives values of p.

Original entry on oeis.org

7, 13, 31, 91, 451, 539, 1397, 1417, 1777, 3139, 14419, 39751, 77507, 96157, 158173, 214979, 263113, 496723, 1046179, 3415297, 3546371, 4306549, 9456677, 18338311, 45521269, 243377803, 766707661, 1023263789, 8032822531, 92635306249, 151318414531, 352799777983

Offset: 1

Hugo Pfoertner, Sep 16 2011

nonn

2*(Positions of records in A195352) + 1.

Checked up to n = 10^13.

			a(3)=31 because it is the first number for which the representation n=2*p+q needs a larger value of p than for all smaller odd numbers. 31=2*7+17, whereas all smaller odd n can be expressed using p=2 or p=3.

See A002091.

Hugo Pfoertner, Table of n, a(n) for n = 1..41

Cf. A195352, A195354 (records of p), A194828 (similar, but looking for records of q with p maximized)

a(36)-a(41) from Hugo Pfoertner, Sep 26 2011

A194831 Records in the number of ways to express an odd number as a sum 2*p+q, with p, q primes.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 16, 21, 22, 26, 27, 31, 32, 35, 43, 48, 52, 65, 77, 87, 101, 104, 115, 128, 133, 146, 155, 169, 180, 188, 194, 196, 201, 209, 225, 228, 248, 250, 282, 286, 325, 332, 359, 391, 400, 443, 449, 470, 555, 579, 582, 679, 741

Offset: 1

Hugo Pfoertner, Sep 12 2011

nonn

Records in A046927. The growth rate of this sequence makes the slow growth of A194829 plausible, i.e. 2*n+1 can be represented by 2*p+q with q<

			a(1)=1: A194830(1)=7 has 1 representation 7=2*2+3; a(2)=2 representations of A194830(2)=9=2*2+5=2*3+3; a(3)=4 representations of A194830(3)=17=2*2+13=2*3+11=2*5+7=2*7+3.

See A046927.

Hugo Pfoertner, Table of n, a(n) for n = 1..322

Cf. A194830 [record-setting numbers], A046927, A194828, A194829.

ways[n_] := ways[n] = (w = 0; Do[ p = 2k + 1; q = n - k; If[PrimeQ[p] && PrimeQ[q], w++], {k, 1, n}]; w); record = 0; A194831 = Reap[Do[If[ways[n] > record, record = ways[n]; Print["2n+1 = ", 2n + 1, " record = ", record]; Sow[{ways[n], n}]], {n, 0, 12000}]][[2, 1]][[All, 1]] (* Jean-François Alcover, Dec 05 2012 *)

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

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A002092 From a Goldbach conjecture: records in A185091.

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A195353 Odd numbers n>5 in the representation n=2*p+q, p, q prime, q maximal, at which a larger p than for any smaller n is needed. A195354 gives values of p.

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A194831 Records in the number of ways to express an odd number as a sum 2*p+q, with p, q primes.

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