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.
f[n_,x_]:=n*x+x*(x+1)/2;Table[f[Prime[n],Prime[n+2]-Prime[n]-1]-Prime[n+1],{n,5!}]
f[n_,x_]:=n*x+x*(x+1)/2;Select[Table[f[Prime[n],Prime[n+2]-Prime[n]-1]-Prime[n+1],{n,6!}],PrimeQ[ # ]&]
Select[Table[Total[Select[Range[Prime[n],Prime[n+2]],CompositeQ]],{n,1000}],PrimeQ] (* Harvey P. Dale, May 13 2017 *)
46 is in the sequence because the fourth set of consecutive natural numbers that are also nonprimes is {8, 9, 10}, the sum of divisors of 8 is 1+2+4+8=15, the sum of divisors of 9 is 1+3+9=13, the sum of divisors of 10 is 1+2+5+10=18, so a(4) = 15+13+18 = 46.
with(numtheory): a:= n-> `if`(n=1, 1, add(sigma(i), i=ithprime(n)+1..ithprime(n+1)-1)): seq(a(n), n=1..80); # Alois P. Heinz, Oct 18 2011
Table[Plus@@Flatten[Divisors[Range[Prime[n] - (-1)^Prime[n], Prime[n + 1] + (-1)^Prime[n + 1]]]], {n, 2, 50}] (* Alonso del Arte, Oct 18 2011 *)
The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with runs:
{3,5,7}, {11,13}, {17,19}, {23}, {29,31}, {37}, {41,43}, {47}, {53}, ...
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, ...
with runs:
{3}, {2,2}, {1}, {2}, {1}, {2}, {1,1}, {2}, {1}, {2}, {1,1,1,1}, {2,2}, ...
with sums a(n).
Total/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ], #1+2==#2&]//Most]//Most
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