A377295 - OEIS

%I #27 Dec 23 2024 21:58:22

%S -1,-1,139,1279,15319,102199,1011079,10054399,100687891,1000860859,

%T 10004248351,100048116199,1000245990631,10000171206199,

%U 100000029166651,1000000001958499,10000010020185919,100000022659152859,1000000088358667051,10000000476596855539,100000000728055460899

%N a(n) is the least n-digit prime which is the sum of the squares of six consecutive nonnegative numbers, or -1 if no such prime exists.

%C From _Robert Israel_, Dec 23 2024: (Start)

%C a(n) is the first n-digit prime, if any, in A027867.

%C a(n) == 1 or 19 (mod 30) if it is not -1. (End)

%H Robert Israel, <a href="/A377295/b377295.txt">Table of n, a(n) for n = 1..996</a>

%e 139 is the smallest 3-digit prime number that can be expressed as the sum of the squares of six consecutive numbers. Specifically, the sum of the squares of the numbers from 2 to 7 is 139:

%e Sum_{i=1..6} (1+i)^2 = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 4 + 9 + 16 + 25 + 36 + 49 = 139.

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

%p  for k from ceil(sqrt(6*10^(n-1)-105)/6 - 5/2) do

%p    p:= 55 + 30*k + 6*k^2;

%p    if p >= 10^n then return -1 fi;

%p    if isprime(p) then return p fi;

%p   od

%p end proc:

%p f(1):= -1: f(2):= -1:

%p map(f, [$1..25]); # _Robert Israel_, Dec 23 2024

%o (Python)

%o from math import isqrt

%o from sympy import isprime

%o from itertools import count

%o def f(m): return sum((m+i)**2 for i in range(6))

%o def a(n):

%o     b = 10**(n-1)

%o     m = isqrt(b//6) - 5

%o     return next(t for i in count(m) if (t:=f(i)) >= b and isprime(t))

%o print([a(n) for n in range(3, 23)]) # _Michael S. Branicky_, Oct 25 2024

%Y Cf. A027865, A027867, A376992.

%K sign,base

%O 1,3

%A _Jean-Marc Rebert_, Oct 23 2024