A081257 -id:A081257 - 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.

A081256 Greatest prime factor of n^3 + 1.

Original entry on oeis.org

2, 3, 7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 13, 307, 7, 127, 421, 463, 13, 79, 601, 31, 37, 757, 271, 67, 19, 331, 151, 1123, 397, 97, 43, 67, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 61, 181, 43, 2551, 379, 919, 409, 2971, 79, 103, 3307, 163

Offset: 1

Jan Fricke, Mar 14 2003

Record values appear to match the terms of A002383 for n>1. - Bill McEachen, Oct 18 2023

R. J. Mathar and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 5000 terms from R. J. Mathar)
J. Buchmann, K. Győry, M. Mignotte, and N. Tzanakis, Lower bounds for P(x^3+k), an elementary approach, Publ. Math. Debrecen, Vol. 38, No. 1-2 (1991), pp. 145-163.

Cf. A081257, A081258, A096173, A096174.

Cf. A006530, A001093, A002383, A014442, A096172.

A081256 := proc(n)
    A006530(n^3+1) ;
end proc:
seq(A081256(n),n=1..20) ; # R. J. Mathar, Feb 13 2014

Table[Max[Transpose[FactorInteger[n^3 + 1]][[1]]], {n, 25}]

a(n)=my(f=factor(n^3+1)); f[#f~,1] \\ Charles R Greathouse IV, Mar 08 2017

A081256(n)=vecmax(factor(n^3+1)[,1]) \\ It seems slightly slower to get the last element using ...[-1..-1][1]. - M. F. Hasler, Jun 15 2018

a(n) = A006530(A001093(n)). - M. F. Hasler, Jun 13 2018

a(n) >= 31 for n >= 70 (Buchmann et al., 1991). - Amiram Eldar, Oct 25 2024

More terms from Harvey P. Dale, Mar 22 2003

More terms from Hugo Pfoertner, Jun 20 2004

A096175 Numbers k such that k^3-1 is an odd semiprime.

Original entry on oeis.org

6, 8, 12, 14, 20, 24, 38, 54, 62, 80, 90, 110, 138, 150, 164, 168, 192, 194, 272, 278, 314, 332, 348, 398, 402, 434, 500, 572, 642, 644, 720, 728, 762, 798, 812, 860, 864, 878, 920, 992, 1020, 1022, 1070, 1092, 1098, 1118, 1130, 1182, 1202, 1230, 1260, 1308

Offset: 1

Hugo Pfoertner, Jun 22 2004

nonn

			a(1)=6 because 6^3 - 1 = 216 - 1 = 215 = 5*43.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Dario A. Alpern, Factorization using the Elliptic Curve Method

Cf. A096173: k^3+1 is an odd semiprime; A081257: largest prime factor of k^3-1; A096176 (k^3-1)/(k-1) is prime; A046315.

forstep (k=2,1310,2,if(bigomega(k^3-1)==2,print1(k,", ")))
\\ Hugo Pfoertner, Nov 28 2017

A256148 Primitive prime factors of the cyclotomic polynomial sequence Phi(3, k) (or Phi(6, k)) in the order in which they occur.

Original entry on oeis.org

3, 7, 13, 31, 43, 19, 73, 37, 157, 61, 211, 241, 307, 127, 421, 463, 79, 601, 757, 271, 67, 331, 151, 1123, 397, 97, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 181, 2551, 379, 919, 409, 2971, 3307, 163, 3541, 523, 3907, 613, 4423, 4831, 1657, 5113, 751

Offset: 1

Robert Price, Mar 16 2015

Phi(3,k) = k^2 + k + 1 and Phi(6,k) = k^2 - k + 1.
Interesting scatter plot.
The terms correspond to the new primes of A081257 in the order of their appearance for n>1 and when A081257(m)>m. - Bill McEachen, Oct 13 2022

Robert Price, Table of n, a(n) for n = 1..733
Bernhard Helmes, Prime sieving on the polynomial f(n)=n^2+n+1.

Cf. A005529, A248874, A256144, A256145, A256146, A081257.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[6, k]], prim]], {k, 1000}]; prim

lista(nn) = {vs = []; for (n=1, nn, vp = factor(polcyclo(6,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

A081258 Numbers k > 1 such that k^3 - 1 (or equivalently k^2 + k + 1) has no prime factor greater than k.

Original entry on oeis.org

16, 18, 22, 30, 49, 67, 68, 74, 79, 81, 87, 100, 102, 121, 135, 137, 146, 149, 154, 158, 159, 163, 165, 169, 172, 178, 181, 191, 211, 221, 229, 230, 235, 256, 262, 263, 269, 273, 277, 291, 292, 301, 305, 313, 315, 324, 326, 334, 352, 361, 372, 373, 380, 393

Offset: 1

Jan Fricke, Mar 14 2003

One might also include 1 as a term here. - R. J. Mathar, Oct 11 2011

			16 is a term: 16^3 - 1 = 4095 = 3*3*5*7*13.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A081256, A081257.

isA081258 := proc(n)
        numtheory[factorset](n^3-1) ;
        if max(op(%)) <= n then
                true;
        else
                false;
        end if;
end proc;
for n from 1 to 400 do
        if isA081258(n) then
                printf("%d,",n);
        end if;
end do: # R. J. Mathar, Oct 11 2011

Select[Range[2, 1000], FactorInteger[#^3 - 1][[-1, 1]] <= #&] (* Jean-François Alcover, Jun 15 2020 *)

Name changed by Robert Israel, Nov 11 2016

A096176 Numbers k such that (k^3-1)/(k-1) is prime.

Original entry on oeis.org

2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218

Offset: 1

Hugo Pfoertner, Jun 22 2004

nonn

Numbers k > 1 such that k^2 + k + 1 is a prime. - Vincenzo Librandi, Nov 16 2010

Therefore essentially the same as A002384. - Georg Fischer, Oct 06 2018

			a(5) = 8 because (8^3-1)/(8-1) = 511/7 = 73 is prime.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A096174 (n^3+1)/(n+1) is prime, A081257 largest prime factor of n^3-1, A096175 n^3-1 is an odd semiprime.
Cf. A028491, A004061. - Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009
Cf. A002384.

[n: n in [2..220] |IsPrime((n^3-1) div (n -1))]; // Vincenzo Librandi, Oct 07 2018

Select[Range[2,550],PrimeQ[(#^3-1)/(#-1)]&] (* Harvey P. Dale, Sep 10 2009 *)

is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jun 05 2017

3 and 5 added by Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009

Corrected terms, including many previously omitted terms, from Harvey P. Dale, Sep 10 2009

A321069 Greatest prime factor of n^3+2.

Original entry on oeis.org

3, 5, 29, 11, 127, 109, 23, 257, 43, 167, 43, 173, 733, 1373, 307, 683, 983, 2917, 2287, 4001, 157, 71, 283, 223, 5209, 47, 127, 3659, 24391, 587, 9931, 113, 433, 6551, 809, 569, 307, 27437, 433, 10667, 439, 239, 1559, 223, 91127, 16223, 4153, 457, 39217, 62501

Offset: 1

Charles R Greathouse IV, Oct 27 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. R. Heath-Brown, The largest prime factor of x^3+2, Proceedings of the London Mathematical Society, 82:3 (2000), pp. 554-596.
Christopher Hooley, On the greatest prime factor of a cubic polynomial, Journal für die reine und angewandte Mathematik, 303 (1978), pp. 21-50.
A. J. Irving, The largest prime factor of x^3+2, arXiv:1412.0024 [math.NT], 2014.

Greatest prime factors of polynomials: A006530 (n), A076565 (2n+1), A076566 (3n+3), A076567 (4n+6), A164314 (n^2-2), A076605 (n^2-1), A014442 (n^2+1), A069902 (n^2+n), A074399 (n^2+n), A199423 (2n^2+n), A089619 (2n^2+2n+1), A037464 (4n^2-1), A253254 (9n^2-7n), A093074 (n^3-n), A081257 (n^3-1), A081256 (n^3+1), A321069(n^3+2), A281793 (n^3+n^2+n+1), A281793 (n^4-1), A096172 (n^4+1), A190136 (n^4 + 6n^3 + 11n^2 + 6n), A140538 (2n^4+1), A240548 (n^5+1), A281794 (n^5+n^3+n^2+1), A240549 (n^6+1), A240550 (n^7+1), A240551 (n^8+1), A240552 (n^9+1), A240553 (n^10+1).

[Maximum(PrimeDivisors(n^3 + 2)): n in [1..60]]; // Vincenzo Librandi, Oct 27 2018

Table[FactorInteger[n^3 + 2] [[-1, 1]], {n, 80}] (* Vincenzo Librandi, Oct 27 2018 *)

a(n) = vecmax(factor(n^3+2)[,1]); \\ Michel Marcus, Oct 27 2018

cp's OEIS Frontend

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

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A081256 Greatest prime factor of n^3 + 1.

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A096175 Numbers k such that k^3-1 is an odd semiprime.

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A256148 Primitive prime factors of the cyclotomic polynomial sequence Phi(3, k) (or Phi(6, k)) in the order in which they occur.

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A081258 Numbers k > 1 such that k^3 - 1 (or equivalently k^2 + k + 1) has no prime factor greater than k.

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A096176 Numbers k such that (k^3-1)/(k-1) is prime.

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A321069 Greatest prime factor of n^3+2.

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