A002091 -id:A002091 - cp's OEIS frontend

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.

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.

A219252 Smallest prime q such that 2n+1 = p + 4q for some odd prime p, otherwise 0 if no such q exists.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 2, 2, 3, 3, 2, 7, 2, 2, 3, 2, 5, 3, 2, 5, 3, 2, 2, 3, 3, 2, 0, 2, 2, 3, 3, 2, 7, 2, 5, 3, 2, 5, 3, 5, 2, 7, 2, 2, 3, 2, 2, 3, 2, 5, 3, 5, 5, 7, 5, 2, 7, 2, 7, 3, 2, 2, 3, 3, 11, 7, 2, 2, 3, 3, 2, 7, 3, 2, 19, 2, 5, 3, 2

Offset: 1

Michel Lagneau, Apr 11 2013

nonn

a(38) = 0.

Conjecture: except m = 77, all odd number > 9 are of the form m = p + 4*q where p and q are prime numbers.

			3 + 4*2 = 11 => a(5) = 2;
5 + 4*2 = 13 => a(6) = 2;
7 + 4*2 = 15 => a(7) = 2;
5 + 4*3 = 17 => a(8) = 3.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A195352, A002091, A219254.

for n from 11 by 2 to 200 do:jj:=0:for j from 1 to 1000 while (jj=0) do:q:=ithprime(j):p:=n-4*q:if p> 0 and type(p,prime)=true  then jj:=1:printf(`%d, `,q):else fi:od:if jj=0 then printf(`%d, `,0):else fi:od:

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

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

A195352 Smallest prime p such that 2n+1 = 2p + q for some odd prime q.

Original entry on oeis.org

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

Offset: 3

Hugo Pfoertner, Sep 16 2011

nonn

Related to Lemoine's conjecture, which states that all odd integers > 5 can be represented as 2*p+q, p, q primes.

			a(3)=2 because 2*3+1=7=2*2+3; a(4)=2: 2*3+1=9=2*2+5; a(5)=2: 11=2*2+7; a(6)=3: 13=2*3+7.

See A002091.

Hugo Pfoertner, Table of n, a(n) for n = 3..10000
Wikipedia, Lemoine's conjecture

Cf. A002091, A103506 (smallest q), A046927, A195353 (positions of records), A195354 (records), A103153 (prime p=2 excluded), A185091.

spp[n_]:=Module[{p=2},While[CompositeQ[(2n+1)-2p],p=NextPrime[p]];p]; Array[ spp,90,3] (* Harvey P. Dale, Jun 02 2022 *)

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]];

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

A195354 Records of primes p in the representation of odd n>5 by n=2*p+q, p, q prime, q maximal. A195353 gives the values of n at which an increase of p is required.

Original entry on oeis.org

2, 3, 7, 19, 31, 41, 47, 67, 79, 139, 181, 229, 233, 277, 307, 383, 421, 463, 619, 643, 659, 691, 743, 967, 1231, 1483, 1609, 1931, 2389, 2719, 2791, 2953, 2971, 3079, 3121, 3217, 3301, 3319, 3617, 3719, 3767

Offset: 1

Hugo Pfoertner, Sep 16 2011

nonn

Values of records in A195352. Related to Lemoine's conjecture. Due to the large number of possible representations (A046927) it is both possible to represent n=2*p+q with q<

A194829) and with p<

Checked up to n=10^13.

See A002091.

Cf. A195352, A195353, A194829.

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

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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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A219252 Smallest prime q such that 2*n+1 = p + 4*q for some odd prime p, otherwise 0 if no such q exists.

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Maple

A002092 From a Goldbach conjecture: records in A185091.

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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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A195352 Smallest prime p such that 2*n+1 = 2*p + q for some odd prime q.

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Mathematica

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

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Magma

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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A195354 Records of primes p in the representation of odd n>5 by n=2*p+q, p, q prime, q maximal. A195353 gives the values of n at which an increase of p is required.

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A219252 Smallest prime q such that 2n+1 = p + 4q for some odd prime p, otherwise 0 if no such q exists.

A195352 Smallest prime p such that 2n+1 = 2p + q for some odd prime q.