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A222532 a(1)=2; for n >= 1, a(n+1) is the least prime p_m such that a(n)=p_m-p_{m-1}+...+(-1)^{m-k}p_k for some 0

%I A222532 #24 Apr 18 2013 02:58:39
%S A222532 2,5,7,13,17,23,31,37,43,53,59,67,73,83,89,101,109,113,131,149,157,
%T A222532 163,173,179,197,223,257,263,269,277,283,311,347,389,401,421,431,487,
%U A222532 503,523,557,569,577,601,613,641,661,709,733,739,773,823,827,857,883,929,947,953,977,983,997,1009,1019,1031,1039,1051,1097,1117,1129,1151,1181,1223,1229,1237,1249,1279,1327,1361,1373,1423,1459,1481,1499,1543,1559,1571,1601,1621,1627,1669,1693,1699,1721,1733,1759,1783,1823,1873,1973,2011
%N A222532 a(1)=2; for n >= 1, a(n+1) is the least prime p_m such that a(n)=p_m-p_{m-1}+...+(-1)^{m-k}p_k for some 0<k<m.
%C A222532 Conjecture: For any given prime p, if we define b(1)=p and let b(n+1) be the least prime p_m such that b(n)=p_m-p_{m-1}+...+(-1)^{m-k}p_k for some 0<k<m, then a(n)=b(n') for some positive integers n and n'. In other words, if we take all the primes as vertices of a simple graph T and let two vertices p and q>p adjacent if and only if q is the least prime p_m such that p=p_m-p_{m-1}+...+(-1)^{m-k}p_k for some 0<k<m, then the graph T is a tree!
%C A222532 Clearly the graph T contains no cycle. The vertices on the unique path connecting 2 and 71 are listed (in order) below: 2, 5, 7, 13, 17, 23, 31, 37, 43, 53, 59, 67, 73, 83, 89, 101, 109, 113, 131, 149, 139, 107, 97, 79, 71.
%H A222532 Zhi-Wei Sun, <a href="/A222532/b222532.txt">Table of n, a(n) for n = 1..10000</a>
%H A222532 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), no.8, 2794-2812.
%e A222532 a(2)=5 and a(3)=7 since 2=5-3 and 5=7-5+3.
%t A222532 k=1
%t A222532 n=1
%t A222532 s[0_]:=0
%t A222532 s[n_]:=s[n]=Prime[n]-s[n-1]
%t A222532 Do[If[m==1,Print[n," ",2]];If[m==k,n=n+1;Do[If[s[j]-(-1)^(j-i)*s[i]==Prime[m],k=j;Print[n," ",Prime[j]];Goto[aa]],{j,m+1,PrimePi[3Prime[m]]},{i,0,j-2}]];
%t A222532 Label[aa];Continue,{m,1,1000}]
%Y A222532 Cf. A000040, A008347, A222566, A163846, A163847.
%K A222532 nonn
%O A222532 1,1
%A A222532 _Zhi-Wei Sun_, Feb 24 2013