A359437 - OEIS

%I #36 May 07 2024 02:00:20

%S 2,17,11,7,239,167,1933,9241,19319,120121,649991,4564559,513239,

%T 11324041,31831799,54708721,59219161,215975759,241431959,265012441,

%U 549789239,138389159,3336693359,1990674841

%N a(n) is the first prime p such that there are exactly n numbers i with 1 <= i < p such that one of i*p-(p-i) and i*p+(p-i) is a prime and the other is the square of a prime.

%C Suggested in an email from _J. M. Bergot_.

%C It appears that in most cases the squares are either all i*p-(p-i) or all i*p+(p-i). However, this is not the case for a(3) or a(18).

%C a(21) = 138389159.

%e a(1) = 17 because 2*17 - (17 - 2) =  19, 2*17 + (17 - 2) = 7^2.

%e a(2) = 11 because 3*11 - (11 - 3) = 5^2, 3*11 + (11 - 3) =  41;

%e                   5*11 - (11 - 5) = 7^2, 5*11 + (11 - 5) =  61.

%e a(3) =  7 because 2*7  -  (7 - 2) = 3^2, 2*7  +  (7 - 2) =  19;

%e                   3*7  -  (7 - 3) =  17, 3*7  +  (7 - 3) = 5^2;

%e                   4*7  -  (7 - 4) = 5^2, 4*7  +  (7 - 4) =  31.

%p f:= proc(p) local x, S1, S2, R1, R2;

%p S1:= {msolve(x^2 = -p, p+1)};

%p R1:= select(i -> i < p and isprime(i*p+p-i), map(t -> (t^2+p)/(p+1), select(isprime, map(rhs@op,S1))));

%p S2:= {msolve(x^2 = p, p-1)};

%p R2:= select(i -> i < p and isprime(i*p-p+i), map(t -> (t^2-p)/(p-1), select(isprime, map(rhs@op,S2))));

%p nops(R1 union R2)

%p end proc:

%p f(2):= 0:

%p V:= Array(0..12): count:= 0:

%p p:= 1:

%p while count < 13 do

%p p:= nextprime(p); v:= f(p);

%p if V[v] = 0 then V[v]:= p; count:= count+1 fi

%p od:

%p convert(V,list);

%o (Python)

%o from sympy import sqrt_mod_iter, nextprime, isprime

%o def A359437(n):

%o     p = 1

%o     while (p:=nextprime(p)):

%o         if len(set(filter(lambda x:isprime(p*(x+1)-x),((d**2+p)//(p+1) for d in sqrt_mod_iter(-p,p+1) if isprime(d)))) | set(filter(lambda x: isprime(p*(x-1)+x),((d**2-p)//(p-1) for d in sqrt_mod_iter(p,p-1) if isprime(d)))))==n:

%o             return p # _Chai Wah Wu_, May 06 2024

%K nonn,more

%O 0,1

%A _Robert Israel_, Dec 31 2022

%E a(21) from _Robert Israel_, Dec 31 2022

%E a(20), a(22)-a(23) from _Chai Wah Wu_, May 06 2024