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.
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
Keywords
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
Keywords
Comments
Related to Lemoine's conjecture, similar to A002091, but with q=1 excluded. Checked up to 2*10^13.
Examples
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.
References
- See A002091.
Links
- Hugo Pfoertner, Table of n, a(n) for n = 1..50
- See A002091.
A195352 Smallest prime p such that 2*n+1 = 2*p + q for some odd prime q.
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
Keywords
Comments
Related to Lemoine's conjecture, which states that all odd integers > 5 can be represented as 2*p+q, p, q primes.
Examples
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.
References
- See A002091.
Links
- Hugo Pfoertner, Table of n, a(n) for n = 3..10000
- Wikipedia, Lemoine's conjecture
Crossrefs
Programs
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Mathematica
spp[n_]:=Module[{p=2},While[CompositeQ[(2n+1)-2p],p=NextPrime[p]];p]; Array[ spp,90,3] (* Harvey P. Dale, Jun 02 2022 *)
Comments
A194829) and with p<
References
Crossrefs
Extensions