A343496 First point of the straight lines in A340649.

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

Offset: 1

Author

Simon Strandgaard and Jamie Morken, Apr 17 2021

Keywords

• nonn
• more

Comments

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.

Examples

			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.
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.
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.
		

Links

• Simon Strandgaard, Visualization of the first 5 terms.

Crossrefs

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

Programs

Ruby

n = 1
last_prime = 2
find_gap = 2
result = []
Prime.each(10_000) do |prime|
    next if prime == 2
    gap = prime - last_prime
    if gap == find_gap
        value = (n * prime) % last_prime
        if value == n * gap
            result << n
            find_gap += 2
        end
    end
    n += 1
    last_prime = prime
end
p result

Formula

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