A228466 Smallest odd number expressible in exactly n ways as p + 2*m^2 where p is 1 or a prime and m >= 0.
5777, 1, 3, 13, 19, 55, 61, 169, 139, 271, 181, 391, 439, 559, 661, 619, 829, 859, 1069, 1081, 1459, 1489, 1609, 1741, 1951, 2029, 2509, 2341, 3631, 3769, 3331, 3961, 4525, 4189, 4261, 5281, 4801, 6229, 6361, 5911, 6439, 7111, 6319, 13081, 9931, 8869, 10321
Offset: 0
Keywords
Examples
a(3) = 13 = 5+2*2^2 = 11+2*1^2 = 13+2*0^2. 13 is the smallest odd number expressible in exactly 3 ways. a(4) = 19 = 1+2*3^2 = 11+2*2^2 = 17+2*1^2 = 19+2*0^2. 19 is the smallest odd number expressible in exactly 4 ways. a(5) = 55 = 5+2*5^2 = 23+2*4^2 = 37+2*3^2 = 47+2*2^2 = 53+2*1^2. 55 is the smallest odd number expressible in exactly 5 ways.
Links
- Donovan Johnson, Table of n, a(n) for n = 0..10000
- G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
- L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., Vol. 66, No. 1 (1993), 45-47.
- M. Stern, Sur une assertion de Goldbach relative aux nombres impairs, Nouvelles Annales Math., 15 (1856) pp. 23-24.
- Index entries for sequences related to Goldbach conjecture
Programs
-
Mathematica
(* finds terms < mx *) upto[mx_] := Block[{r = Floor[1+mx/2], k, t, p, s = {}}, t = 0*Range@r; p = Prime@ Range@ PrimePi@ mx; p[[1]] = 1; t[[# + Range[0, Sqrt[r - #]]^2]]++ & /@ ((1 + p)/2); k = 0; While[(r = Position[t, k, 1, 1]) != {}, k++; AppendTo[s, 2 r[[1,1]] - 1]]; s]; upto[10^5] (* Giovanni Resta, Aug 23 2013 *) -
PARI
/* finds terms up to a(1000) */ mx=10602619; v=vector(mx); nn=vector(1000); p=vector(701940); p[1]=1; pr=2; for(j=2, 701940, pr=nextprime(pr+1); p[j]=pr); for(m=0, 2302, m2=2*m^2; for(j=1, 701940, s=m2+p[j]; if(s<=mx, v[s]++, next(2)))); forstep(j=1, mx, 2, if(v[j]==0, write("b228466.txt", 0 " " j); j=mx)); forstep(j=1, mx, 2, if(v[j]>0, if(v[j]<=1000, if(nn[v[j]]==0, nn[v[j]]=j)))); for(n=1, 1000, write("b228466.txt", n " " nn[n]))