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A222532
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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.
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7
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2, 5, 7, 13, 17, 23, 31, 37, 43, 53, 59, 67, 73, 83, 89, 101, 109, 113, 131, 149, 157, 163, 173, 179, 197, 223, 257, 263, 269, 277, 283, 311, 347, 389, 401, 421, 431, 487, 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
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OFFSET
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1,1
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COMMENTS
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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!
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.
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LINKS
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EXAMPLE
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a(2)=5 and a(3)=7 since 2=5-3 and 5=7-5+3.
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MATHEMATICA
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k=1
n=1
s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
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}]];
Label[aa]; Continue, {m, 1, 1000}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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