A350868 - OEIS

%I #26 Jan 31 2022 06:46:20

%S 2,3,29,569,6701,64919,1720289,256828391,33090566651,248804328761,

%T 55130906480861,119321483551349

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

%C 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.

%C a(12) > 10^15. - _Martin Ehrenstein_, Jan 31 2022

%e 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.

%p P:= select(isprime, [2,seq(i,i=3..2*10^6,2)]):

%p f:= proc(n) local k;

%p      for k from 1 do

%p        if P[n+k] <> P[n]+2*k^2 then return k-1 fi

%p      od

%p end proc:

%p V:= Array(0..6):

%p for n from 1 to nops(P)-21 do

%p   v:= H(n);

%p   if V[v] = 0 then V[v]:= P[n] fi;

%p od:

%p convert(V,list);

%o (Python)

%o from sympy import prime, nextprime

%o def A350868(n):

%o     if n < 2:

%o         return 2+n

%o     qlist = [prime(i)-2 for i in range(2,n+2)]

%o     p = prime(n+1)

%o     mlist = [2*k**2 for k in range(1,n+1)]

%o     while True:

%o         if qlist == mlist:

%o             return p-mlist[-1]

%o         qlist = [q-qlist[0] for q in qlist[1:]]

%o         r = nextprime(p)

%o         qlist.append(r-p+qlist[-1])

%o         p = r # _Chai Wah Wu_, Jan 24 2022

%Y Cf. A212769.

%K nonn,more

%O 0,1

%A _J. M. Bergot_ and _Robert Israel_, Jan 20 2022

%E a(7) from _David A. Corneth_, Jan 20 2022

%E a(8) from _Chai Wah Wu_, Jan 25 2022

%E a(9) from _Martin Ehrenstein_, Jan 26 2022

%E a(10)-a(11) from _Martin Ehrenstein_, Jan 31 2022