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A363702
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Let m be the least integer for which there exists a strictly increasing sequence u of n integers in {1..m} such that x = (2 * Sum_{k=1..n} k*prime(u(k))) / (n*(n+1)) is an integer. a(n) is the least x, or -1 if no such integer x exists.
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0
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2, 4, 11, 12, 12, 12, 16, 21, 24, 22, 24, 31, 32, 34, 41, 40, 42, 44, 49, 50, 52, 52, 61, 63, 62, 68, 70, 75, 74, 82, 88, 89, 92, 92, 102, 106, 106, 106, 113, 118, 118, 118, 125, 127, 132, 132, 141, 148, 142, 150, 154, 158, 158, 162, 171, 175, 172, 178, 181, 187
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OFFSET
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1,1
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LINKS
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EXAMPLE
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1*prime(1) + 2*prime(3) = 12, 1 + 2 = 3 and 12/3 = 4 is an integer and no other strictly increasing sequence of 2 primes <= prime(3) gives a lesser result, so a(2) = 4.
1*prime(3) + 2*prime(5) + 3*prime(6) = 5 + 2*11 + 3*13 = 66, 66/6 = 11 is an integer and no other strictly increasing sequence of 3 primes <= prime(6) gives a lesser result, so a(3) = 11.
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PROG
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(PARI) is(u)={my(s=0, c=0, n=#u, sc=n*(n+1)/2); for(i=1, n, my(p=prime(u[i])); s+=i*p); s%sc==0}
f(u)={my(s=0, n=#u, vc=vector(n, x, x), sc=n*(n+1)/2, v=[]); if(is(u), for(i=1, #u, v=concat(v, prime(u[i]))); s=v*vc~; return(s/sc)); -1}
find(m=n, n)={my(x=m, sol=[], solmin=-1); forsubset([m, n], p, my(vp=Vec(p)); if(is(vp), my(x=f(vp)); if(solmin==-1, solmin=x); if(solmin>0&&x<solmin, solmin=x)); if(x<vecmin(vp), break)); return(solmin)}
a(n)={my(m=n); x=find(m, n); while(x==-1, m++; x=find(m, n)); return(x)}
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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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