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%I #15 Mar 13 2023 04:44:00
%S 3,5,7,11,13,19,29,41,43,59,83,89,107,109,127,139,157,163,173,199,211,
%T 223,257,271,277,293,307,331,347,367,397,421,443,457,491,541,557,587,
%U 601,631,691,761,769,821,911,941,971,991,1009,1033,1103,1129,1153,1201
%N Primes in "Ulam's Prime sequence". A prime is in the sequence iff p+1 can be expressed in exactly 1 way as the sum of 2 previous distinct primes.
%C a(1) = 3, a(2) = 5; for n >= 3, a(n) is smallest prime which is uniquely a(j) + a(k) - 1, with 1<= j < k < n.
%C Is the (3,5) sequence finite or infinite? Note that (3,7) as a starting sequence has only 2 terms and (7,11) yields 7, 11, 17, 23, 29 only. Equally using -1 as a rule creates more variants.
%C The sequence continues at least up to a(2227) = 400031.
%C After about 500 terms, the graph of this sequences appears almost linear. - _T. D. Noe_, Jan 20 2008
%H T. D. Noe, <a href="/A078425/b078425.txt">Table of n, a(n) for n=1..10000</a>
%H <a href="/index/U#Ulam_num">Index entries for Ulam numbers</a>
%e a(3)=7 as 8=3+5. a(4)=11 as 12=5+7 (and nothing else).
%o (PARI) v=vector(1220);vc=2;v[1]=3;v[2]=5; forprime (p=7,1220,p1=p+1;pc=0;fl=0;for (i=1,vc-1, for (j=i+1,vc,if (v[i]+v[j]==p1,pc++);if (pc>1,fl=1);if (fl,break));if (fl,break));if (pc==0,fl=1);if (!fl,vc++;v[vc]=p));print(vecextract(v,concat("1..",vc)))
%Y Cf. A002858 (Ulam numbers), A002859, A003666, A003667, A001857, A048951, A007300.
%K nonn
%O 1,1
%A _Jon Perry_, Dec 29 2002
%E Edited and extended by _Klaus Brockhaus_, Apr 14 2005
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