A024012 -id:A024012 - cp's OEIS frontend

A174120 Partial sums of A024012.

1, 2, 2, 1, 1, 8, 36, 115, 307, 738, 1662, 3589, 7541, 15564, 31752, 64295, 129575, 260358, 522178, 1046105, 2094281, 4190992, 8384812, 16772891, 33549531, 67103338, 134211526, 268428525, 536863197, 1073733268, 2147474192

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 08 2010

A024012 2^n-n^2 -> 1,1,0,-1,0,7,28,79,192,431,924,1927,3952,..

Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

f[n_]:=Sum[2^i-i^2,{i,0,n}];Table[f[n],{n,0,5!}]

Vec(-x*(x^3+4*x^2-4*x+1)/((x-1)^4*(2*x-1)) + O(x^100)) \\ Colin Barker, Oct 27 2014

a(n) = (-1+2^n-n/6+n^2/2-n^3/3). G.f.: -x*(x^3+4*x^2-4*x+1) / ((x-1)^4*(2*x-1)). - Colin Barker, Oct 27 2014

A072180 Numbers k such that 2^k - k^2 is prime.

Original entry on oeis.org

5, 7, 9, 17, 19, 51, 53, 81, 83, 119, 189, 219, 227, 301, 455, 461, 623, 2037, 2221, 2455, 3547, 5515, 6825, 8303, 9029, 12103, 49989, 55525, 64773, 80307, 119087, 141915, 192023, 205933, 301683, 307407

Offset: 1

Daniel Gronau (Daniel.Gronau(AT)gmx.de), Jun 30 2002

The numbers corresponding to k = 2037, 2221, 3547 and 5515 have been certified prime with Primo. - Rick L. Shepherd, Nov 10 2002
The remaining k's > 1000 correspond only to probable primes.
Certainly k must be odd. Let N(k) = 2^k - k^2. Additional restrictions come from the facts that 7 | N(k) if k is in {2, 4, 5, 6, 10, 15} mod 21 and 17 | N(k) if k is in {31, 57, 61, 71, 107, 109, 113, 131} mod 136. - Daniel Gronau, Jul 06 2002
Henri Lifchitz found the terms > 40000 in 2001 and 119087 in March 2002. - Hugo Pfoertner, Nov 16 2004

Henri Lifchitz, Renaud Lifchitz, PRP Top Records 2^n-n^2.

Cf. A024012, A064539, A075896, A072164.

Do[ If[ PrimeQ[ 2^n - n^2], Print[n]], {n, 1, 22850, 2}]

is(n)=isprime(2^n-n^2) \\ Charles R Greathouse IV, Feb 17 2017

Edited and extended by Robert G. Wilson v, Jul 01 2002
More terms from Hugo Pfoertner, Nov 16 2004
More terms from Henri Lifchitz submitted by Ray Chandler, Mar 02 2007

A024040 a(n) = 4^n - n^4.

Original entry on oeis.org

1, 3, 0, -17, 0, 399, 2800, 13983, 61440, 255583, 1038576, 4179663, 16756480, 67080303, 268397040, 1073691199, 4294901760, 17179785663, 68719371760, 274877776623, 1099511467776, 4398046316623, 17592185810160, 70368743897823, 281474976378880, 1125899906451999

Offset: 0

N. J. A. Sloane

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-30,50,-45,21,-4).

Cf. A024012, A024026, A058794. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[4^n-n^4: n in [0..30]]; // Vincenzo Librandi, May 14 2011

seq(4^n - n^4, n=0..50); # Robert Israel, Dec 29 2014

lst={}; Do[AppendTo[lst,4^n-n^4],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[4^n-n^4,{n,0,40}] (* Harvey P. Dale, Nov 10 2019 *)

G.f.: (1-6*x+3*x^2+23*x^3+48*x^4+3*x^5)/((1-4*x)*(1-x)^5).

E.g.f.: exp(4*x)-(x^4+6*x^3+7*x^2+x)*exp(x). - Robert Israel, Dec 29 2014

A024054 a(n) = 5^n - n^5.

Original entry on oeis.org

1, 4, -7, -118, -399, 0, 7849, 61318, 357857, 1894076, 9665625, 48667074, 243891793, 1220331832, 6102977801, 30516818750, 152586842049, 762938033268, 3814695376057, 19073483852026, 95367428440625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11, -45, 95, -115, 81, -31, 5).

Cf. A024012, A024026, A058794, A024040. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[5^n-n^5: n in [0..30]]; // Vincenzo Librandi, May 14 2011

lst={}; Do[AppendTo[lst,5^n-n^5],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[5^n-n^5,{n,0,20}] (* or *) LinearRecurrence[{11,-45,95,-115,81,-31,5},{1,4,-7,-118,-399,0,7849},30] (* Harvey P. Dale, Oct 15 2014 *)

a(n)=5^n-n^5 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (-6*x^6 - 123*x^5 - 319*x^4 - 44*x^3 + 6*x^2 + 7*x - 1)/((x - 1)^6*(5*x - 1)). - Harvey P. Dale, Oct 15 2014

a(0)=1, a(1)=4, a(2)=-7, a(3)=-118, a(4)=-399, a(5)=0, a(6)=7849, a(n) = 11*a(n-1) - 45*a(n-2) + 95*a(n-3) - 115*a(n-4) + 81*a(n-5) - 31*a(n-6) + 5*a(n-7). - Harvey P. Dale, Oct 15 2014

A024068 a(n) = 6^n - n^6.

Original entry on oeis.org

1, 5, -28, -513, -2800, -7849, 0, 162287, 1417472, 9546255, 59466176, 361025495, 2173796352, 13055867207, 78356634560, 470173593951, 2821093130240, 16926635307167, 101559922656192, 609359692964615, 3656158376062976, 21936950554611735, 131621703728887232, 789730222905566927

Offset: 0

N. J. A. Sloane

6^n in the formula can be removed (for example) with the following Maple code: "with(gfun): rec1:={u1(0)=1,u1(n+1)=6*u1(n)}: rec2:={u2(n)=n^6}: poltorec(u1(n)-u2(n),[rec1,rec2],u1(n),u2(n)],a(n));". This yields a polynomial recurrence: {a(n+1)-5*n^6+6*n^5+15*n^4+20*n^3+15*n^2-6*a(n)+6*n+1, a(0) = 1} that can further be transformed into a linear recurrence with constant coefficients. - Georg Fischer, Feb 23 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (13,-63,161,-245,231,-133,43,-6).

Cf. A024012, A024026, A058794, A024040, A024054. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[6^n-n^6: n in [0..30]]; // Vincenzo Librandi, May 14 2011

Table[6^n-n^6,{n,0,20}] (* Harvey P. Dale, Jan 30 2019 *)

From Chai Wah Wu, Jan 26 2020: (Start)
a(n) = 13*a(n-1) - 63*a(n-2) + 161*a(n-3) - 245*a(n-4) + 231*a(n-5) - 133*a(n-6) + 43*a(n-7) - 6*a(n-8) for n > 7.
G.f.: (5*x^7 + 348*x^6 + 1734*x^5 + 1545*x^4 + 5*x^3 - 30*x^2 - 8*x + 1)/((x - 1)^7*(6*x - 1)). (End)

More terms from Georg Fischer, Feb 23 2021

A075896 Primes of the form 2^k - k^2.

Original entry on oeis.org

7, 79, 431, 130783, 523927, 2251799813682647, 9007199254738183, 2417851639229258349405791, 9671406556917033397642519, 664613997892457936451903530140158127, 784637716923335095479473677900958302012794430558004278391

Offset: 1

Zak Seidov, Oct 17 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..20

Primes in A024012.

Cf. A061119, A072180.

[a: n in [1..200] | IsPrime(a) where a is 2^n-n^2]; // Vincenzo Librandi, Dec 07 2011

Select[Table[2^n - n^2, {n, 500}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

for(n=2,10^7,if(isprime(2^n-n^2),print1(2^n-n^2",")))

a(n) = A024012(A072180(n)). - Elmo R. Oliveira, Feb 18 2025

More terms from Ralf Stephan, Mar 30 2003

A024082 7^n-n^7.

Original entry on oeis.org

1, 6, -79, -1844, -13983, -61318, -162287, 0, 3667649, 35570638, 272475249, 1957839572, 13805455393, 96826261890, 678117659345, 4747390650568, 33232662134145, 232630103648534, 1628412985690417, 11398894291501404, 79792265017612001, 558545862282195466

Offset: 0

N. J. A. Sloane

sign

a(20)=79792265017612001 and a(24)=191581231375979942977 are primes, thus terms of A123206. - M. F. Hasler, Aug 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A024012, A024026, A058794, A024040, A024054, A024068.

[7^n-n^7: n in [0..25]]; // Vincenzo Librandi, May 16 2011

Table[7^n - n^7, {n, 0, 25}] (* Stefan Steinerberger, Apr 08 2006 *)

G.f.: (1-9*x-85*x^2-407*x^3+5991*x^4+15665*x^5+8245*x^6+831*x^7+8*x^8)/((1-7*x)*(1-x)^8). - Bruno Berselli, May 16 2011

More terms from Stefan Steinerberger, Apr 08 2006

A024025 a(n) = 3^n - n^2.

Original entry on oeis.org

1, 2, 5, 18, 65, 218, 693, 2138, 6497, 19602, 58949, 177026, 531297, 1594154, 4782773, 14348682, 43046465, 129139874, 387420165, 1162261106, 3486784001, 10460352762, 31381059125, 94143178298, 282429535905, 847288608818

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Cf. sequences of the form k^n-n^2: A024012 (k=2), this sequence (k=3), A024038 (k=4), A024051 (k=5), A024064 (k=6), A024077 (k=7), A024090 (k=8), A024103 (k=9), A024116 (k=10), A024129 (k=11), A024142 (k=12).

Cf. A000244, A000290.

[3^n-n^2: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

A024025:=n->3^n-n^2: seq(A024025(n), n=0..50); # Wesley Ivan Hurt, Jan 11 2017

Table[3^n - n^2, {n, 0, 25}] (* or *) CoefficientList[Series[(1 - 4 x + 5 x^2 + 2 x^3)/((1 - 3 x) (1 - x)^3), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 05 2014 *)

[3^n-n^2 for n in range(31)] # G. C. Greubel, Aug 18 2023

G.f.: (1-4*x+5*x^2+2*x^3)/((1-3*x)*(1-x)^3). - Vincenzo Librandi, Oct 05 2014
a(n) = 6*a(n-1) -12*a(n-2) +10*a(n-3) -3*a(n-4) for n>3. - Vincenzo Librandi, Oct 05 2014
a(n) = A000244(n) - A000290(n). - Michel Marcus, Oct 05 2014
E.g.f.: exp(3*x) - x*(1+x)*exp(x). - G. C. Greubel, Aug 18 2023

A024096 a(n) = 8^n - n^8.

Original entry on oeis.org

1, 7, -192, -6049, -61440, -357857, -1417472, -3667649, 0, 91171007, 973741824, 8375575711, 68289495040, 548940083167, 4396570722048, 35181809198207, 281470681743360, 2251792837927807, 18014387489521408, 144115171092292831

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (17, -108, 372, -798, 1134, -1092, 708, -297, 73, -8).

Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[8^n-n^8: n in [0..25]]; // Vincenzo Librandi, May 16 2011

Table[8^n - n^8, {n, 0, 20}] (* or *) LinearRecurrence[ {17, -108, 372, -798, 1134, -1092, 708, -297, 73, -8}, {1, 7, -192, -6049, -61440, -357857, -1417472, -3667649, 0, 91171007}, 20] (* Harvey P. Dale, Oct 10 2013 *)

a(n)=8^n-n^8 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1 - 10*x - 203*x^2 - 2401*x^3 + 18851*x^4 + 109207*x^5 + 120743*x^6 + 34061*x^7 + 1984*x^8 + 7*x^9) / ((1-8*x)*(1-x)^9). - Bruno Berselli, May 16 2011

a(0)=1, a(1)=7, a(2)=-192, a(3)=-6049, a(4)=-61440, a(5)=-357857, a(6)=-1417472, a(7)=-3667649, a(8)=0, a(9)=91171007; for n>9, a(n) = 17*a(n-1) - 108*a(n-2) + 372*a(n-3) - 798*a(n-4) + 1134*a(n-5) - 1092*a(n-6) + 708*a(n-7) - 297*a(n-8) + 73*a(n-9) - 8*a(n-10). - Harvey P. Dale, Oct 10 2013

A024110 a(n) = 9^n - n^9.

Original entry on oeis.org

1, 8, -431, -18954, -255583, -1894076, -9546255, -35570638, -91171007, 0, 2486784401, 29023111918, 277269756129, 2531261328956, 22856131408177, 205852688735274, 1852951469375105, 16677063111790072, 150094436937708753

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082, A024096. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[9^n-n^9: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011

lst={}; Do[AppendTo[lst,9^n-n^9],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[9^n-n^9,{n,0,20}] (* Harvey P. Dale, Jul 27 2022 *)

From Chai Wah Wu, Jan 26 2020: (Start)
a(n) = 19*a(n-1) - 135*a(n-2) + 525*a(n-3) - 1290*a(n-4) + 2142*a(n-5) - 2478*a(n-6) + 2010*a(n-7) - 1125*a(n-8) + 415*a(n-9) - 91*a(n-10) + 9*a(n-11) for n > 10.
G.f.: (-10*x^10 - 4507*x^9 - 131015*x^8 - 779378*x^7 - 1317686*x^6 - 637664*x^5 - 43448*x^4 + 10210*x^3 + 448*x^2 + 11*x - 1)/((x - 1)^10*(9*x - 1)). (End)

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A174120 Partial sums of A024012.

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A072180 Numbers k such that 2^k - k^2 is prime.

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A024040 a(n) = 4^n - n^4.

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A024054 a(n) = 5^n - n^5.

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A024068 a(n) = 6^n - n^6.

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A075896 Primes of the form 2^k - k^2.

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A024082 7^n-n^7.

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