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

0,1

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

Links

Table of n, a(n) for n=0..11.

Example

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.

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);

Prog

(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

Crossrefs

Cf. A212769.

Sequence in context: A064893 A141514 A078727 * A270550 A366477 A076977

Adjacent sequences: A350865 A350866 A350867 * A350869 A350870 A350871

Keyword

nonn,more

Author

J. M. Bergot and Robert Israel, Jan 20 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

Status

approved