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%I #19 Aug 05 2023 06:25:29
%S 5777,1,3,13,19,55,61,169,139,271,181,391,439,559,661,619,829,859,
%T 1069,1081,1459,1489,1609,1741,1951,2029,2509,2341,3631,3769,3331,
%U 3961,4525,4189,4261,5281,4801,6229,6361,5911,6439,7111,6319,13081,9931,8869,10321
%N Smallest odd number expressible in exactly n ways as p + 2*m^2 where p is 1 or a prime and m >= 0.
%H Donovan Johnson, <a href="/A228466/b228466.txt">Table of n, a(n) for n = 0..10000</a>
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
%H L. Hodges, <a href="https://www.jstor.org/stable/2690477">A lesser-known Goldbach conjecture</a>, Math. Mag., Vol. 66, No. 1 (1993), 45-47.
%H M. Stern, <a href="http://www.numdam.org/item/NAM_1856_1_15__23_0/">Sur une assertion de Goldbach relative aux nombres impairs</a>, Nouvelles Annales Math., 15 (1856) pp. 23-24.
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e 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.
%e 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.
%e 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.
%t (* 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 *)
%o (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]))
%Y Cf. A007697, A016067, A046921.
%K nonn
%O 0,1
%A _Donovan Johnson_, Aug 22 2013
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