A343496 - OEIS

%I #25 May 23 2021 03:08:43

%S 5,31,194,1061,6456,40080,251721,1617206,10553419,69709769,465769825

%N First point of the straight lines in A340649.

%C prime(a(n)+1) - prime(a(n)) = n*2. E.g., for n=4: prime(a(4)+1) - prime(a(4)) = 4*2, prime(1062) - prime(1061) = 4*2, 8521 - 8513 = 8.

%H Simon Strandgaard, <a href="/A343496/a343496.png">Visualization of the first 5 terms</a>.

%F a(n) = smallest k that satisfies A001223(k) = 2*n and A340649(k) = A141042(k).

%e For n=1, consider k's with prime gap 1*2 = 2, i.e., k's such that A001223(k)=2. k=5 is the first place where A001223(k)=2 and A141042(k)=A340649(k), so a(1)=5.

%e For n=2, consider k's with prime gap 2*2 = 4, i.e., k's such that A001223(k)=4. k=31 is the first place where A001223(k)=4 and A141042(k)=A340649(k), so a(2)=31.

%e For n=3, consider k's with prime gap 3*2 = 6, i.e., k's such that A001223(k)=6. k=194 is the first place where A001223(k)=6 and A141042(k)=A340649(k), so a(3)=194.

%o (Ruby) n = 1

%o last_prime = 2

%o find_gap = 2

%o result = []

%o Prime.each(10_000) do |prime|

%o     next if prime == 2

%o     gap = prime - last_prime

%o     if gap == find_gap

%o         value = (n * prime) % last_prime

%o         if value == n * gap

%o             result << n

%o             find_gap += 2

%o         end

%o     end

%o     n += 1

%o     last_prime = prime

%o end

%o p result

%Y Cf. A000040, A001223, A141042, A340649, A029707, A029709, A320701, A320702, A320703.

%K nonn,more

%O 1,1

%A _Simon Strandgaard_ and _Jamie Morken_, Apr 17 2021