A140719 -id:A140719 - cp's OEIS frontend

A153257 a(n) = n^3 - (n+1)^2.

-1, -3, -1, 11, 39, 89, 167, 279, 431, 629, 879, 1187, 1559, 2001, 2519, 3119, 3807, 4589, 5471, 6459, 7559, 8777, 10119, 11591, 13199, 14949, 16847, 18899, 21111, 23489, 26039, 28767, 31679, 34781, 38079, 41579, 45287, 49209, 53351, 57719, 62319, 67157, 72239

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 22 2008

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A045991, A069778, A081437, A140719.

Table[n^3-(n+1)^2,{n,0,40}] (* Harvey P. Dale, Oct 05 2022 *)

my(x='x+O('x^43)); Vec((x^3+5*x^2+x-1)/(x-1)^4) \\ Elmo R. Oliveira, Aug 27 2025

From Elmo R. Oliveira, Aug 27 2025: (Start)
G.f.: (-1 + x + 5*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-1 + x)*(1 + 3*x + x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Elmo R. Oliveira, Aug 27 2025

A154111 Numbers n such that (n+1)^2 - n^3 is a (positive or negative) prime.

Original entry on oeis.org

1, 3, 5, 6, 8, 11, 12, 15, 18, 20, 27, 33, 35, 39, 41, 45, 48, 50, 54, 65, 66, 68, 86, 87, 92, 96, 99, 107, 116, 122, 123, 126, 138, 140, 149, 150, 156, 159, 161, 164, 165, 167, 170, 177, 182, 185, 188, 191, 192, 198, 200, 207, 209, 219, 228, 237, 239, 240, 242, 252

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

G. C. Greubel, Table of n, a(n) for n = 1..2000

Cf. A140719, A005097.

[n: n in [0..500] | IsPrime((n+1)^2-n^3)]; // Vincenzo Librandi, Nov 26 2010

Select[Range[300],PrimeQ[(#+1)^2-#^3]&] (* Harvey P. Dale, Dec 06 2015 *)

is(n)=isprime(abs((n+1)^2 - n^3)) \\ Charles R Greathouse IV, Sep 02 2016

A154112 Numbers k such that (k+1)^3 - k^2 is prime.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 16, 19, 20, 32, 34, 37, 42, 44, 45, 49, 51, 52, 56, 60, 62, 70, 72, 74, 75, 77, 79, 81, 89, 90, 95, 96, 97, 100, 101, 104, 105, 111, 114, 115, 121, 126, 131, 136, 145, 151, 154, 156, 161, 171, 174, 175, 180, 182, 191, 199, 200, 202, 207

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

			2^3-1^2 = 7, 3^3-2^2 = 23, ...

G. C. Greubel, Table of n, a(n) for n = 1..2000

Cf. A140719, A005097, A154111.

[n: n in [0..5000] | IsPrime(((n+1)^3-n^2))]; // Vincenzo Librandi, Nov 26 2010

a[n_]:=(n+1)^3-n^2;lst={};Do[If[PrimeQ[a[n]],AppendTo[lst,n]],{n,6!}];lst
Select[Range[210],PrimeQ[(#+1)^3-#^2]&] (* Harvey P. Dale, Sep 20 2011 *)

is(n)=isprime((n+1)^3 - n^2) \\ Charles R Greathouse IV, Sep 02 2016

A154113 Primes of the form (n+1)^3 - n^2.

Original entry on oeis.org

7, 23, 109, 191, 307, 463, 919, 1231, 1607, 2053, 4657, 7639, 8861, 34913, 41719, 53503, 77743, 89189, 95311, 122599, 138007, 146173, 182057, 223381, 246203, 353011, 383833, 416399, 433351, 468623, 505759, 544807, 721079, 745471, 875711

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

			2^3-1^2=7, 3^3-2^2=23, ...

G. C. Greubel, Table of n, a(n) for n = 1..2000

Cf. A140719, A005097, A154111, A154112.

[a: n in [0..100] | IsPrime(a) where a is (n+1)^3 - n^2]; // Vincenzo Librandi, Sep 02 2016

a[n_]:=(n+1)^3-n^2;lst={};Do[If[PrimeQ[a[n]],AppendTo[lst,a[n]]],{n,6!}];lst
Select[(#+1)^3-#^2&/@Range[200],PrimeQ]  (* Harvey P. Dale, Mar 14 2011 *)

for(n=1,1e3, if(isprime(t=(n+1)^3 - n^2), print1(t", "))) \\ Charles R Greathouse IV, Sep 02 2016

A154115 Numbers n such that n + 3 is prime.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 16, 20, 26, 28, 34, 38, 40, 44, 50, 56, 58, 64, 68, 70, 76, 80, 86, 94, 98, 100, 104, 106, 110, 124, 128, 134, 136, 146, 148, 154, 160, 164, 170, 176, 178, 188, 190, 194, 196, 208, 220, 224, 226, 230, 236, 238, 248, 254, 260, 266, 268, 274, 278

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

			a(2) = 2 since (2 + 2)^2 - (2 + 1)^2 - 2 = 5.

G. C. Greubel, Table of n, a(n) for n = 1..2000

Cf. A140719, A005097, A154111, A154112, A154113.

Cf. A067076 (a(n-1)/2).

[n: n in [0..500] | IsPrime((n+2)^2-(n+1)^2-n)]; // Vincenzo Librandi, Nov 26 2010

A154115 := proc(n) ithprime(n+1)-3 ; end proc: # R. J. Mathar, May 09 2010

a[n_]:=(n+2)^2-(n+1)^2-n;lst={};Do[If[PrimeQ[a[n]],AppendTo[lst,n]],{n,6!}];lst
Select[Range[0,300],PrimeQ[(#+2)^2-(#+1)^2-#]&] (* Harvey P. Dale, Nov 06 2013 *)
Prime[Range[2,100]]-3 (* Harvey P. Dale, Jul 15 2017 *)

is(n)=isprime(n+3) \\ Charles R Greathouse IV, Sep 02 2016

a(n) = A086801(n+1). - R. J. Mathar, May 09 2010

New name based on a comment by Franklin T. Adams-Watters, Jan 30 2009

A171771 Primes of form n^6-(n+1)^5.

Original entry on oeis.org

971, 431441, 838949, 2614691, 6770161, 43845881, 570523321, 9244951889, 33640090481, 41402933641, 81303824909, 126165366289, 137240997911, 346860978491, 372445245449, 525200678549, 726938163649, 774170449439

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Dec 18 2009

nonn

(1) It is conjectured that sequence is infinite.

(2) p=97=prime(25) is the smallest prime such that (p-1)^6-p^5 and p^6-(p+1)^5 are primes.

			4^6-5^5=971 and 9^6-10^5=431441 are prime.

Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
Derrick H. Lehmer, Guide to Tables in the Theory of Numbers Washington, D.C. 1941

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002327, A140719, A087191

Select[Table[n^6-(n+1)^5,{n,3,100}],PrimeQ] (* Harvey P. Dale, Mar 06 2019 *)

Edited by D. S. McNeil, Nov 21 2010

A172192 Numbers k such that k^6 - (k+1)^5 is prime.

Original entry on oeis.org

4, 9, 10, 12, 14, 19, 29, 46, 57, 59, 66, 71, 72, 84, 85, 90, 95, 96, 97, 114, 119, 122, 155, 157, 190, 191, 204, 207, 212, 221, 222, 244, 251, 256, 276, 285, 286, 289, 294, 300, 301, 307, 319, 320, 337, 344, 355, 359, 380, 382, 392, 400, 411, 422, 426, 441, 451

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jan 29 2010

Corresponding primes are in A171771. Negative values of primes are obtained for 1 and 2.

			4^6-(4+1)^5 = 971 is prime, so 4 is in the sequence.
5^6-(5+1)^5 = 7849 = 47*167 is composite, so 5 is not in the sequence.
9^6-(9+1)^5 = 431441 is prime, so 9 is in the sequence.

Cf. A171771, A002327 (primes of form n^2-n-1), A140719 (primes of form n^3-(n+1)^2), A087191 (primes of form n^4-(n+1)^3).

[ n: n in [1..460] | IsPrime(p) and p gt 0 where p is n^6-(n+1)^5 ];

Select[Range[3,500],PrimeQ[#^6-(#+1)^5]&] (* Harvey P. Dale, Apr 25 2011 *)

is(n)=isprime(n^6-(n+1)^5) \\ Charles R Greathouse IV, Jun 13 2017

Edited, extended, non-specific references removed and MAGMA program added by Associate Editors OEIS, Mar 05 2010

cp's OEIS Frontend

A153257 a(n) = n^3 - (n+1)^2.

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A154111 Numbers n such that (n+1)^2 - n^3 is a (positive or negative) prime.

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A154112 Numbers k such that (k+1)^3 - k^2 is prime.

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A154113 Primes of the form (n+1)^3 - n^2.

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A154115 Numbers n such that n + 3 is prime.

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A171771 Primes of form n^6-(n+1)^5.

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A172192 Numbers k such that k^6 - (k+1)^5 is prime.

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Magma

Mathematica