A076565 -id:A076565 - cp's OEIS frontend

A316859 Triangle read by rows constructed from A076565 as sum of greatest prime factors.

6, 8, 8, 10, 10, 10, 6, 12, 12, 6, 14, 8, 14, 8, 14, 16, 16, 10, 10, 16, 16, 8, 18, 18, 6, 18, 18, 8, 20, 10, 20, 14, 14, 20, 10, 20, 22, 22, 12, 16, 22, 16, 12, 22, 22, 10, 24, 24, 8, 24, 24, 8, 24, 24, 10, 26, 12, 26, 20, 16, 26, 16, 20, 26, 12, 26, 8, 28, 14, 22, 28, 18, 18, 28, 22, 14, 28, 8

Offset: 1

Fred Daniel Kline, Jul 15 2018

The greatest number in row k is 2*k + 4, thus consecutive rows identify consecutive even numbers (sums of primes).
To get the n-th row: copy (1...n) of A076565, reverse, and add together.
When primes meet primes we get the maximum values. When primes or prime factors meet prime factors, we get lesser values. (Spot checked. Still empirical.)

			{ 6},           <--- copy (1,1) of A076565, add together
{ 8,  8},       <--- copy (1,2) of A076565, reverse, and add together
{10, 10, 10},   <--- copy (1,3) of A076565, reverse, and add together
{ 6, 12, 12,  6},
{14,  8, 14,  8, 14},
{16, 16, 10, 10, 16, 16},
{ 8, 18, 18,  6, 18, 18,  8}, <=== differences with A316858 begin here
{20, 10, 20, 14, 14, 20, 10, 20},
{22, 22, 12, 16, 22, 16, 12, 22, 22},
{10, 24, 24,  8, 24, 24,  8, 24, 24, 10},
{26, 12, 26, 20, 16, 26, 16, 20, 26, 12, 26},
{ 8, 28, 14, 22, 28, 18, 18, 28, 22, 14, 28,  8}

Cf. A076565, A316858 related triangle using lpfs.

gpf[n_] := FactorInteger[2 n + 1][[-1, 1]]; A076565 = Array[gpf, 12];
a = Table[A076565[[1 ;; -n]] + Reverse[A076565[[1 ;; -n]]],
  {n, Length[A076565], 1, -1}];

A076567 Greatest prime divisor of 4n+6 (sum of four successive integers).

Original entry on oeis.org

5, 7, 3, 11, 13, 5, 17, 19, 7, 23, 5, 3, 29, 31, 11, 7, 37, 13, 41, 43, 5, 47, 7, 17, 53, 11, 19, 59, 61, 7, 13, 67, 23, 71, 73, 5, 11, 79, 3, 83, 17, 29, 89, 13, 31, 19, 97, 11, 101, 103, 7, 107, 109, 37, 113, 23, 13, 17, 11, 41, 5, 127, 43, 131, 19, 5, 137, 139, 47, 13

Offset: 1

Zak Seidov, Oct 19 2002

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

Cf. A006530, A076565.

f:= n -> max(numtheory:-factorset(n)):
map(f, [seq(i,i=10..1000,4)]); # Robert Israel, Nov 10 2020

FactorInteger[#][[-1,1]]&/@(4*Range[70]+6) (* Harvey P. Dale, Mar 12 2017 *)

a(n) = {f = factor(4*n+6); return (f[length(f~), 1]);} \\ Michel Marcus, Apr 19 2013

a(n) = vecmax(factor(4*n+6)[,1]); \\ Michel Marcus, Nov 10 2020

a(n) = A076565(n+1) = A006530(4n+6). - R. J. Mathar, Jul 31 2008

A140538 Greatest prime factor of 2*n^4 + 1.

Original entry on oeis.org

3, 11, 163, 19, 139, 2593, 1601, 2731, 1193, 113, 227, 619, 577, 8537, 73, 43691, 55681, 209953, 307, 9697, 388963, 52057, 1091, 337, 260417, 304651, 3011, 4937, 471521, 1620001, 691, 5419, 32491, 46889, 90947, 25643, 11057, 15619, 7499, 7793

Offset: 1

Donald S. McDonald, Jul 06 2008

			a(2)= 11 because 2*2^4+1 = 33 = 3*11, greatest prime factor is 11.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

[#f eq 0 select 1 else f[ #f][1] where f is Factorization(2*n^4 + 1): n in [1..50]]; // Vincenzo Librandi, Feb 17 2015

Table[Max[Transpose[FactorInteger[2 n^4 + 1]][[1]]], {n, 50}] (* Vincenzo Librandi, Feb 17 2015 *)

a(n) = f = factor(2*n^4+1); f[#f~,1]; \\ Michel Marcus, Aug 28 2013

a(n) = A006530(2n^4+1) = A076565(n^4). - R. J. Mathar, Aug 04 2008

Extended from a(16) on by R. J. Mathar, Aug 04 2008

A179485 Sums of two successive primes s such that s+-3 are primes.

Original entry on oeis.org

8, 100, 1120, 1220, 1300, 2240, 2380, 2414, 3536, 3634, 4906, 4940, 5566, 5740, 6706, 7240, 8864, 9224, 9394, 10136, 10850, 12040, 12476, 12586, 12920, 13180, 13334, 13754, 14630, 14720, 15134, 16270, 17710, 18430, 18800, 19916, 21014, 21320

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 16 2010

nonn

Intersection of A001043 and A087695. - Robert Israel, Oct 25 2017

			3+5=8,8-3=5(prime),8+3=11(prime),..

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

Cf. A074924, A074925, A167597, A176984, A176986, A075593, A075594, A116360, A076565

Cf. A001043, A087695.

q:= 2; p:= 3;
count:= 0:
while count < 100 do
  q:= p; p:= nextprime(p);
  s:= q+p;
  if isprime(s-3) and isprime(s+3) then
    count:= count+1; A[count]:= s;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Oct 25 2017

q=3;Select[Table[Prime[n]+Prime[n+1],{n,7!}],PrimeQ[ #-q]&&PrimeQ[ #+q]&]
Select[Total/@Partition[Prime[Range[1400]],2,1],AllTrue[#+{3,-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2018 *)

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

A380609 Primes a single step away from a cycle under the mapping p-> gpf(2*p+1).

Original entry on oeis.org

2, 17, 31, 37, 67, 71, 73, 97, 103, 137, 149, 157, 181, 199, 211, 227, 241, 269, 283, 313, 337, 367, 379, 409, 487, 541, 563, 577, 587, 607, 617, 643, 661, 769, 787, 857, 877, 907, 929, 937, 977, 997, 1039, 1093, 1151, 1187, 1237, 1453, 1543, 1567, 1579, 1621

Offset: 1

Johannes M.V.A. Koelman, Jan 28 2025

nonn

The cycle that gets entered consists of the primes in A287865. It appears that the mapping p -> gpf(2*p+1) produces no other cycles.

Conjecture: under repeated mapping all primes ultimately enter the same cycle.

			Prime 2 is in the sequence as it maps to 5. And so is 17 as it maps to 7.  The primes 3, 5, 7, 11, 13, 19, 23 and 47 are not included, as they are part of the cycle itself (and hence considered zero iterations away from the cycle).

Cf. A006530, A076565, A287865.

gpf:= n -> max(numtheory:-factorset(n)):
filter:= proc(n) local S,t,x;
  t:= gpf(2*n+1);
  if t = n then return false fi;
  S:= {n,t};
  x:= t;
  do
    x:= gpf(2*x+1);
    if member(x,S) then return (x = t) fi;
    S:= S union {x};
  od;
end proc:
select(filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Feb 03 2025

cp's OEIS Frontend

A316859 Triangle read by rows constructed from A076565 as sum of greatest prime factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A076567 Greatest prime divisor of 4n+6 (sum of four successive integers).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A140538 Greatest prime factor of 2*n^4 + 1.

Views

Author

Keywords

Examples

Links

Programs

Magma

Mathematica

PARI

Formula

Extensions

A179485 Sums of two successive primes s such that s+-3 are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A321069 Greatest prime factor of n^3+2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A380609 Primes a single step away from a cycle under the mapping p-> gpf(2*p+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple