A256148 -id:A256148 - cp's OEIS frontend

A357127 a(n) = A081257(n) if A081257(n) > n, otherwise a(n) = 1.

7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 1, 307, 1, 127, 421, 463, 1, 79, 601, 31, 37, 757, 271, 67, 1, 331, 151, 1123, 397, 97, 43, 67, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 61, 181, 1, 2551, 379, 919, 409, 2971, 79, 103, 3307, 163, 3541, 523, 97, 3907, 109, 73, 613

Offset: 2

Mohammed Bouras, Sep 13 2022

nonn

All the primes in this sequence appear exactly twice.

The new primes encountered seem to match the terms of A256148 for n>1. Bill McEachen, Oct 13 2022

			a(2) = a(a(2) - 2 - 1) = a(7 - 2 - 1) = a(4).
a(3) = a(9) = 3 + 9 + 1 = 13.
a(5) = a(25) = gcd(5^2 + 5 + 1, 25^2 + 25 + 1) = 31.

Cf. A081257, A081256, A108768, A256148.

from sympy import primefactors
def A357127(n): return m if (m:=max(primefactors(n*(n+1)+1))) > n else 1 # Chai Wah Wu, Oct 15 2022

Conjecture 1: If a(n) != 1, then a(n) = a(a(n) - n - 1).
Conjecture 2: If n != m and a(n) = a(m), then
a(n) = gcd(n^2 + n + 1, m^2 + m + 1) = n + m + 1.

A108768 Primes that appear in the sequence p:=x^2+x+1, sieved with a quadratic sieve construction.

Original entry on oeis.org

3, 7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 307, 127, 421, 463, 79, 601, 31, 37, 757, 271, 67, 331, 151, 1123, 397, 97, 43, 67, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 61, 181, 2551, 379, 919, 409, 2971, 79, 103, 3307, 163, 3541, 523, 97, 3907

Offset: 1

Bernhard Helmes (pi(AT)devalco.de), Jun 24 2005

This sequence appears in a website available on web.archive (see Quadratic Sieve Construction link). There is a single appearance of the first term 3, while all other primes appear twice. See A256148 for a version of the sequence consistent with the current version of the website where each prime appears only once. - Ray Chandler, Jul 05 2015

Bernhard Helmes, Prime sieving on the polynomial f(n)=n^2+n+1.
Bernhard Helmes, Quadratic Sieve Construction (web.archive).

Cf. A002061, A002383, A007645, A162471, A256148.

// from Quadratic Sieve Construction link.
liste_max:=10000;
for x from 1 to liste_max do
    liste_x[x]:=x^2+x+1;
    liste_prim[x]:=1;
end_for;
x:=1;
while (x1) then
     print ("Prim ", p, "x = ", x, isprime (p)) ;
     // Aussiebung
     while (stelleRay Chandler, Jul 05 2015

A256153 Primitive prime factors of the cyclotomic polynomial sequence Phi(5,k) in the order in which they occur.

Original entry on oeis.org

5, 31, 11, 71, 311, 2801, 151, 61, 41, 271, 3221, 22621, 30941, 3761, 4931, 88741, 2711, 911, 251, 40841, 245411, 292561, 346201, 521, 8641, 4561, 637421, 732541, 837931, 17351, 601, 1801, 39451, 22571, 49831, 101, 4271, 194681, 191, 401, 2625641, 579281

Offset: 1

Robert Price, Mar 16 2015

nonn

Phi(5,k) = k^4 + k^3 + k^2 + k + 1.

All terms end with the digit 1.

Robert Price, Table of n, a(n) for n = 1..1191

Cf. A005529, A248874, A256144, A256145, A256146, A256148.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[5, k]], prim]], {k, 1000}]; prim

lista(nn) = {vs = []; for (n=1, nn, vp = factor(polcyclo(5,n))[,1]; for (i=1, #vp, if (!vecsearch(vs, vp[i]), print1(vp[i], ", "); vs = vecsort(concat(vs, vp[i]),,8););););} \\ Michel Marcus, Mar 20 2015

cp's OEIS Frontend

A357127 a(n) = A081257(n) if A081257(n) > n, otherwise a(n) = 1.

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A108768 Primes that appear in the sequence p:=x^2+x+1, sieved with a quadratic sieve construction.

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A256153 Primitive prime factors of the cyclotomic polynomial sequence Phi(5,k) in the order in which they occur.

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Mathematica

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