A363702 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.

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

Offset

1,1

Links

Table of n, a(n) for n=1..60.

Example

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.

Prog

(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)}

Crossrefs

Cf. A045345, A050248.

Sequence in context: A218643 A300375 A075488 * A136995 A136993 A136992

Adjacent sequences: A363699 A363700 A363701 * A363703 A363704 A363705

Keyword

nonn

Author

Jean-Marc Rebert, Jun 16 2023

Status

approved