Original entry on oeis.org
4, 3, 13, 13, 513, 801, 6781, 30721, 40513, 1057513, 515313, 16728501, 78402181, 13617661, 472012281, 64846161, 5481873013, 2459693601, 116852093013, 62611784481, 1234170737761, 1565435686113, 17492477581161, 2254878102513, 16836143444113, 229959946206301
Offset: 1
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a[n_]:=(Module[{k=1}, While[!PrimeQ[m=2k^2+2k+1]||IntegerLength[m]
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from math import isqrt
from itertools import count
from sympy import isprime
def A376993(n):
for k in count(isqrt(((a:=10**(n-1))<<1)-1>>2)):
m = 2*k*(k+1)+1
if m >= a and isprime(m):
return m-a # Chai Wah Wu, Oct 13 2024
A377295
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.
Original entry on oeis.org
-1, -1, 139, 1279, 15319, 102199, 1011079, 10054399, 100687891, 1000860859, 10004248351, 100048116199, 1000245990631, 10000171206199, 100000029166651, 1000000001958499, 10000010020185919, 100000022659152859, 1000000088358667051, 10000000476596855539, 100000000728055460899
Offset: 1
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:
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.
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f:= proc(n) local p,k;
for k from ceil(sqrt(6*10^(n-1)-105)/6 - 5/2) do
p:= 55 + 30*k + 6*k^2;
if p >= 10^n then return -1 fi;
if isprime(p) then return p fi;
od
end proc:
f(1):= -1: f(2):= -1:
map(f, [$1..25]); # Robert Israel, Dec 23 2024
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from math import isqrt
from sympy import isprime
from itertools import count
def f(m): return sum((m+i)**2 for i in range(6))
def a(n):
b = 10**(n-1)
m = isqrt(b//6) - 5
return next(t for i in count(m) if (t:=f(i)) >= b and isprime(t))
print([a(n) for n in range(3, 23)]) # Michael S. Branicky, Oct 25 2024
A377294
a(n) is the least n-digit prime which is the sum of the squares of three consecutive numbers, or -1 if no such prime exists.
Original entry on oeis.org
5, 29, 149, 1877, 11909, 100469, 1026677, 10013789, 100676549, 1000611509, 10007613149, 100003082789, 1000092600389, 10000275414869, 100000426365677, 1000004865589109, 10000013191662677, 100000034139489269, 1000000221045632669, 10000000313838962309, 100000002116695737029
Offset: 1
29 is the smallest 2-digit prime number that can be expressed as the sum of the squares of three consecutive numbers. Specifically, the sum of the squares of the numbers from 2 to 4 is 29: Sum_{i=1..3} (1+i)^2 = 2^2 + 3^2 + 4^2 = 4 + 9 + 16 = 29.
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from math import isqrt
from sympy import isprime
from itertools import count
def f(m): return sum((m+i)**2 for i in range(3))
def a(n):
b = 10**(n-1)
m = isqrt(b//3) - 2
m += m&1 # note: m must be even for f(m) to be odd
return next(t for i in count(m, 2) if (t:=f(i)) >= b and isprime(t))
print([a(n) for n in range(2, 22)]) # Michael S. Branicky, Oct 25 2024
Showing 1-3 of 3 results.
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