A350868 a(n) is the first prime p such that the next n primes are p+2*k^2 for k=1..n.

2, 3, 29, 569, 6701, 64919, 1720289, 256828391, 33090566651, 248804328761, 55130906480861, 119321483551349

Offset: 0

Author

J. M. Bergot and Robert Israel, Jan 20 2022

Keywords

• nonn
• more

Comments

If p = prime(m) is a prime such that the next n primes are p+2*k^2 for k=1..n, then A212769(m+k-1) = 2*p+1 for k=1..n.

a(12) > 10^15. - Martin Ehrenstein, Jan 31 2022

Examples

			a(3) = 569 because the next 3 primes after 569 are 571 = 569 + 2*1^2, 577 = 569 + 2*2^2, 587 = 569 + 2*3^2, and 569 is the first prime that works.
		

Crossrefs

Cf. A212769.

Programs

Maple

P:= select(isprime, [2,seq(i,i=3..2*10^6,2)]):
f:= proc(n) local k;
     for k from 1 do
       if P[n+k] <> P[n]+2*k^2 then return k-1 fi
     od
end proc:
V:= Array(0..6):
for n from 1 to nops(P)-21 do
  v:= H(n);
  if V[v] = 0 then V[v]:= P[n] fi;
od:
convert(V,list);

Python

from sympy import prime, nextprime
def A350868(n):
    if n < 2:
        return 2+n
    qlist = [prime(i)-2 for i in range(2,n+2)]
    p = prime(n+1)
    mlist = [2*k**2 for k in range(1,n+1)]
    while True:
        if qlist == mlist:
            return p-mlist[-1]
        qlist = [q-qlist[0] for q in qlist[1:]]
        r = nextprime(p)
        qlist.append(r-p+qlist[-1])
        p = r # Chai Wah Wu, Jan 24 2022

Extensions

a(7) from David A. Corneth, Jan 20 2022

a(8) from Chai Wah Wu, Jan 25 2022

a(9) from Martin Ehrenstein, Jan 26 2022

a(10)-a(11) from Martin Ehrenstein, Jan 31 2022