A081256 -id:A081256 - cp's OEIS frontend

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

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

A240550 Greatest prime factor of n^7+1.

Original entry on oeis.org

2, 43, 547, 113, 449, 197, 911, 5419, 16493, 909091, 1623931, 13063, 22079, 7027567, 10678711, 15790321, 22796593, 32222107, 226871, 10529, 81867661, 86969, 2969, 183458857, 234750601, 59011, 2269, 35771, 574995877, 1118041, 71821, 86171, 219409, 104119, 11831

Offset: 1

T. D. Noe, Apr 07 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A014442, A081256, A096172, A240548-A240553.

Table[FactorInteger[n^7 + 1][[-1, 1]], {n, 100}]

A240551 Greatest prime factor of n^8+1.

Original entry on oeis.org

2, 257, 193, 65537, 11489, 98801, 169553, 673, 21523361, 5882353, 6304673, 260753, 407865361, 16097, 179953, 6700417, 184417, 113607841, 563377, 1505882353, 300673, 3227992561, 623009, 2311681, 29423041, 57734881, 769, 22223646961, 561377, 4855073

Offset: 1

T. D. Noe, Apr 07 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A014442, A081256, A096172, A240548-A240553.

Table[FactorInteger[n^8 + 1][[-1, 1]], {n, 100}]

A240552 Greatest prime factor of n^9+1.

Original entry on oeis.org

2, 19, 37, 109, 5167, 46441, 117307, 87211, 530713, 52579, 590077, 1801, 937, 132049, 811, 38737, 5653, 465841, 236377, 69481, 613, 5966803, 1117, 7561, 6597973, 102966067, 19927, 102547, 10435069, 120871, 1538083, 18837001, 221401, 745903, 612740917, 55117

Offset: 1

T. D. Noe, Apr 07 2014

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A014442, A081256, A096172, A240548, A240549, A240550, A240551, A240553.

Table[FactorInteger[n^9 + 1][[-1, 1]], {n, 100}]

a(n) = vecmax(factor(n^9+1)[,1]); \\ Michel Marcus, Dec 17 2017

A281661 The least common multiple of 1 + n^2 and 1 + n^3.

Original entry on oeis.org

1, 2, 45, 140, 1105, 1638, 8029, 8600, 33345, 29930, 101101, 81252, 250705, 186830, 540765, 381488, 1052929, 712530, 1895725, 1241660, 3208401, 2046902, 5164765, 3224520, 7977025, 4890938, 11899629, 7184660, 17233105, 10268190, 24327901, 14329952, 33588225, 19586210

Offset: 0

R. J. Mathar, Jan 26 2017

If d|(1 + n^2) and d|(1 + n^3), then d|((1 + n^2) - (n*(1 + n^2) - (1 + n^3))^2) = 2*n. If k|n and k|(1 + n^2), then k = 1 is only option since k|n^2 and k|(1 + n^2). So d must be 1 or 2, exactly. Obviously if n is odd, then the greatest d must be 2 since 1 + n^2 and 1 + n^3 are even. If n is even, then d must be 1 since 1 + n^2 and 1 + n^3 are odd.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-15,0,20,0,-15,0,6,0,-1).

Cf. A001093, A002522, A281660.

A281661 := proc(n)
        ilcm(1+n^2,1+n^3);
end proc:

Table[LCM[n^2+1,n^3+1],{n,0,50}] (* Harvey P. Dale, Jun 10 2023 *)

a(n) = lcm(n^2+1, n^3+1); \\ Michel Marcus, Jan 29 2017

a(n) = (n^2 + 1)*(n^3 + 1)/(1 + n%2); \\ Altug Alkan, Jan 29 2017

a(n) = lcm(1+n^2, 1+n^3) = (1+n^2)*(1+n^3)/gcd(1+n^2, 1+n^3).
a(n) = (1+n^2)*(1+n^3)/ A000034(n) with g.f. ( 1 +2*x +39*x^2 +128*x^3 +850*x^4 +828*x^5 +2054*x^6 +832*x^7 +861*x^8 +130*x^9 +35*x^10 ) / ( (x-1)^6 *(1+x)^6 ).
A006530(a(n)) = max( A081256(n), A014442(n)). - R. J. Mathar, Jan 28 2017
a(n) = (3 + (-1)^n)*(1 + n^2 + n^3 + n^5)/4. - Colin Barker, Feb 07 2017

A281794 The largest prime factor of (1+n^2)*(1+n^3).

Original entry on oeis.org

1, 2, 5, 7, 17, 13, 37, 43, 19, 73, 101, 61, 29, 157, 197, 211, 257, 29, 307, 181, 401, 421, 463, 53, 577, 601, 677, 73, 757, 421, 67, 37, 331, 151, 1123, 613, 1297, 137, 67, 1483, 1601, 547, 1723, 139, 631, 1013, 109, 103, 461, 1201, 61, 2551, 541, 919

Offset: 0

R. J. Mathar, Jan 30 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A006530, A014442, A081256, A281661.

[#f eq 0 select 1 else f[ #f][1] where f is Factorization((1+n^2)*(1+n^3)): n in [0..60]]; // Vincenzo Librandi, Jun 03 2017

A281794 := proc(n)
    A006530((1+n^2)*(1+n^3)) ;
end proc:
seq(A281794(n),n=0..60) ;

Table[Max[Transpose[FactorInteger[(1 + n^2) (1 + n^3)]][[1]]], {n, 0, 60}] (* Vincenzo Librandi, Jun 03 2017 *)

a(n) = if (n==0, 1, my(f=factor((1+n^2)*(1+n^3))); vecmax(f[, 1])); \\ Michel Marcus, Jun 03 2017; corrected Jun 13 2022

a(n) = max( A014442(n), A081256(n)).

a(n) = A006530(A281661(n)).

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

Original entry on oeis.org

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.

A240554 Square array of the greatest prime factor of n^k + 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 2, 5, 2, 1, 5, 5, 3, 2, 1, 3, 17, 7, 17, 2, 1, 7, 13, 13, 41, 11, 2, 1, 2, 37, 7, 257, 61, 13, 2, 1, 3, 5, 31, 313, 41, 73, 43, 2, 1, 5, 13, 43, 1297, 521, 241, 547, 257, 2, 1, 11, 41, 19, 1201, 101, 601, 113, 193, 19, 2, 1, 3, 101, 73, 241

Offset: 1

T. D. Noe, Apr 07 2014

T. D. Noe, Rows n = 1..60 of triangle, flattened

Cf. A003992 (n^k), A014442 (k=2), A081256 (k=3), A096172 (k=4).

Cf. A240548-A240553 (k=5 to 10).

Table[FactorInteger[(n-k)^k + 1][[-1,1]], {n, 12}, {k, n}]

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

A243609 Numbers n such that the difference between the greatest prime divisor of n^3 + 1 and the sum of the other distinct prime divisors is equal to +-1.

Original entry on oeis.org

12, 17, 179, 546, 1241, 12520, 19484, 35732, 65933, 76782, 86918, 90035, 94381, 120195, 183677, 209837, 229829, 241951, 288260, 315724, 338712, 344231, 422069, 568346, 597327, 734382, 894504, 1345874, 1635804, 1697093, 2000325, 2043907, 2131745, 2262789, 2492717

Offset: 1

Michel Lagneau, Jun 23 2014

nonn

			12 is in the sequence because 12^3 + 1 = 1729 = 7 * 13 * 19 and 19 - (13+7) = 19 - 20 = -1;
17 is in the sequence because 17^3 + 1 = 4914 = 2*3^3*7*13 and 13 - (7+3+2) = 13 - 12 = 1.

J. Harrington, L. Jones, A. Lamarche, Representing Integers as the Sum of Two Squares in the Ring Z_n, J. Int. Seq. 17 (2014) # 14.7.4.

Cf. A001093, A081256, A244194.

fpdQ[n_]:=Module[{f=Transpose[FactorInteger[n^3+1]][[1]]},Max[f]-Total[Most[f]]==1];gpdQ[n_]:=Module[{g=Transpose[FactorInteger[n^3+1]][[1]]},Max[g]-Total[Most[g]]==-1];Union[Select[Range[2,5*10^6],fpdQ ],Select[Range[2,5*10^6],gpdQ ]]
dgQ[n_]:=Module[{f=FactorInteger[n^3+1][[All,1]],len,a,b},len= Length[ f]-1;{a,b}=TakeDrop[f,len];Abs[Total[a]-b[[1]]]==1]; Select[Range[ 25*10^5],dgQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 03 2019 *)

cp's OEIS Frontend

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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A240550 Greatest prime factor of n^7+1.

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A240551 Greatest prime factor of n^8+1.

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Mathematica

A240552 Greatest prime factor of n^9+1.

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A281661 The least common multiple of 1 + n^2 and 1 + n^3.

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A281794 The largest prime factor of (1+n^2)*(1+n^3).

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

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A240554 Square array of the greatest prime factor of n^k + 1, read by antidiagonals.

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