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%I #41 Feb 13 2024 06:57:24
%S 0,23,53,120,211,1340,1341,1342,1343,1344,2179,3967,15704,15705,16033,
%T 19634,19635,24281,31428,31429,31430,31431,31432,31433,38501,58831,
%U 155964,203713,206699,370310,370311,370312,370313,370314,370315,370316
%N Lonely numbers: distance to closest prime sets a new record.
%H Charles R Greathouse IV and Giovanni Resta, <a href="/A051650/b051650.txt">Table of n, a(n) for n = 0..211</a> (terms < 10^14, first 156 terms from Charles R Greathouse IV)
%e 23 is 4 units away from the closest prime (not including itself), so 23 is in the sequence.
%t d[0] = 2; d[k_] := Min[k - NextPrime[k, -1], NextPrime[k] - k]; a[0] = 0; a[n_] := a[n] = (k = a[n-1] + 1; record = d[a[n-1]]; While[d[k] <= record, k++]; k); Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jan 16 2012 *)
%t dcp[n_]:=Min[n-NextPrime[n,-1],NextPrime[n]-n]; DeleteDuplicates[Table[{n,dcp[n]},{n,0,375000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* _Harvey P. Dale_, Feb 23 2023 *)
%o (PARI) print1(0);w=2;p=2;q=3;forprime(r=5,1e9,if(p+w+w<q,for(t=p+w+1,(q+p)\2,print1(", "t));w=(q-p)\2);t=min(q-p,r-q);if(t>w,w=t;print1(", "q));p=q;q=r) \\ _Charles R Greathouse IV_, Jan 16 2012
%Y Related sequences: A023186-A023188, A046929-A046931, A051650, A051652, A051697-A051702, A051728-A051730.
%Y Distances are in A051730.
%K nonn,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Dec 23 1999 and from _Jud McCranie_, Jun 16 2000
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