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.
s1=s2=0;lst={};Do[p=Prime[n];s1+=p;If[PrimeQ[s1],s2+=s1;If[PrimeQ[s2],AppendTo[lst,s2]]],{n,2,8!}];lst
2 is in the sequence because 2 appears in A007504. 4 is in the sequence because 4 appears in A062198. 5 is in the sequence because 5 appears in A007504. 6 is not in the sequence because 6 is not in A062198. 8 is in the sequence because 8 appears in A086062, 10 is in the sequence because 10 appears in A062198.
alm := proc(n,m) # n-th m-almost prime
option remember;
if n =1 then
2^m ;
else
for a from procname(n-1,m)+1 do
if numtheory[bigomega](a) = m then
return a;
end if;
end do:
end if;
end proc:
almP := proc(n,m) #n-th partial sum of the m-almost primes
add(alm(i,m),i=1..n) ;
end proc:
isA216686 := proc(n) # is n in the sequence?
local m ,k,ps;
m := numtheory[bigomega](n) ;
for k from 1 do
ps := almP(k,m) ;
if ps = n then
return true;
elif ps > n then
return false;
end if;
end do:
end proc:
for n from 1 to 4300 do
if isA216686(n) then
printf("%d,",n) ;
end if;
end do: # R. J. Mathar, Sep 14 2012
is_A216686(n)={ my(m=bigomega(n),t); while(n>0, while(bigomega(t++)!=m,); n-=t); !n} \\ - M. F. Hasler, Sep 23 2012
lista(nn) = {s = 0; forprime(p=2, nn, s += p; if (isprime(s) && isprime(s-2), print1(p, ", ")););} \\ Michel Marcus, Feb 23 2015
k=3: prime(3) = 5 = 2+3 = prime(1) + prime(2). k=7: prime(7) = 17 = 2+3+5+7 = prime(1) + prime(2) + prime(3) + prime(4). k=13: prime(13) = 41 = 2+3+5+7+11+13 = prime(1) + prime(2) + prime(3) + prime(4) + prime(5) + prime(6).
11 is a term because 11 - 2 = 9, 11 - (2 + 3) = 6, 11 - (2 + 3 + 5) = 1, and none of these are prime. 17 is not a term because 17 - (2 + 3 + 5) = 7, which is prime.
primeQ[n_] := n > 0 && PrimeQ[n]; With[{p = Prime[Range[1000]]}, s = Accumulate[p]; q[n_] := AllTrue[s, ! primeQ[n - #] &]; Select[p, q]] (* Amiram Eldar, Dec 04 2022 *)
121 is a term because 121 = 11^2 = 4 + 8 + 9 + 16 + 25 + 27 + 32 = 2^2 + 2^3 + 3^2 + 2^4 + 5^2 + 3^3 + 2^5.
Select[Accumulate[Select[Range[3723875], GCD @@ FactorInteger[#][[All, 2]] > 1 &]], GCD @@ FactorInteger[#][[All, 2]] > 1 &]
37 is a term because 37 is a prime and 37 = 4 + 8 + 9 + 16 = 2^2 + 2^3 + 3^2 + 2^4.
Select[Accumulate[Select[Range[5000000], PrimeOmega[#] > 1 && PrimePowerQ[#] &]], PrimeQ[#] &]
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