A195354 -id:A195354 - cp's OEIS frontend

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.

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

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 *)

cp's OEIS Frontend

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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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 2n+1 = 2p + q for some odd prime q.

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cp's OEIS Frontend

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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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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Author

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Extensions

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

Views

Author

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Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

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