This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(1) = 3 because p(3) + p(4) + p(5) = 5 + 7 + 11 = 23 = A122706(1)^1. a(2) = 6 because p(6) + p(7) + p(8) = 13 + 17 + 19 = 49 = 7^2 = A122706(2)^2.
spQ[n_]:=Module[{n3=n^3,a,b,c,d,e},c=NextPrime[Floor[n3/3]];b=NextPrime[ c,-1];a=NextPrime[b,-1];d=NextPrime[c];e=NextPrime[d];n3==a+b+c || n3==b+c+d || n3==c+d+e];Select[Prime[Range[1200]],spQ] (* Harvey P. Dale, Sep 23 2011 *)
{ p1=prime(1) ; p2=prime(2) ; p3=prime(3) ; n3=p1+p2+p3 ; for(i=1,100000000, if( ispower(n3,3,&n), if(isprime(n), print(n) ) ; ) ; n3 -= p1 ; p1=p2 ; p2=p3 ; p3=nextprime(p3+1) ; n3 += p3 ; ) ; } \\ R. J. Mathar, Jan 13 2007
a(1) does not exist because there is no power of 2 that is a sum of 3 consecutive primes. prime(5)^2 = 11^2 = 121 can be written as 37+41+43, therefore a(5)=2.
{ A123994(n) = my(k,t1,t2,t3,m); k=0; while(1, k++; m=prime(n)^k; t1=precprime(m/3); t2=nextprime(m/3); t3=m-t1-t2; if( ispseudoprime(t3) && ( (t3t2 && t3==nextprime(t2+1)) ), return(k)); ); }
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