A376907 -id:A376907 - cp's OEIS frontend

A376933 a(n) = (A376907(n) - 10^(n-1))/3.

2, 3, 9, 219, 89, 707, 7139, 17397, 127107, 185573, 170099, 8513673, 3730629, 9645947, 20878187, 435128997, 413502597, 7180622897, 28071202809, 4197184407, 53159437779, 72827487477, 408466487673, 1622948986427, 1009480191957, 50924645281527, 141362538039039

Offset: 1

Stefano Spezia, Oct 11 2024

Cf. A011557, A376907.

a[n_]:=(Module[{k=1}, While[!PrimeQ[m=3k^2+3k+1]||IntegerLength[m]

a(n) = my(m=10^(n-1), p); for(k=(sqrtint(12*m-3)-3)\6, oo, p=3*k*(k+1)+1; if(p>m&&isprime(p), return((p-m)/3))); \\ Jinyuan Wang, Mar 01 2025

from itertools import count
from math import isqrt
from sympy import isprime
def A376933(n):
    for k in count(isqrt((((a:=10**(n-1))<<2)-1)//12)):
        m = 3*k*(k+1)+1
        if m >= a and isprime(m):
            return (m-a)//3 # Chai Wah Wu, Oct 13 2024

a(21)-a(27) from Chai Wah Wu, Oct 13 2024

A376992 a(n) is the least n-digit prime of the form j^2 + (j+1)^2.

Original entry on oeis.org

5, 13, 113, 1013, 10513, 100801, 1006781, 10030721, 100040513, 1001057513, 10000515313, 100016728501, 1000078402181, 10000013617661, 100000472012281, 1000000064846161, 10000005481873013, 100000002459693601, 1000000116852093013, 10000000062611784481, 100000001234170737761

Offset: 1

Stefano Spezia, Oct 11 2024

Cf. A001844, A027861, A027862, A376907, A376993.

f:= proc(n) local j,x;
     for j from ceil((sqrt(2*10^(n-1)-1)-1)/2) do
       x:= j^2 + (j+1)^2;
       if isprime(x) then return x fi
     od
end proc:
map(f, [$1..40]); # Robert Israel, Oct 13 2024

a[n_]:=Module[{k=1}, While[!PrimeQ[m=2k^2+2k+1]||IntegerLength[m]

from math import isqrt
from itertools import count
from sympy import prime
def A376992(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 # Chai Wah Wu, Oct 13 2024

Conjecture: a(n+1)/a(n) ~ 10.

cp's OEIS Frontend

A376933 a(n) = (A376907(n) - 10^(n-1))/3.

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