A024082 -id:A024082 - cp's OEIS frontend

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

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)

A024124 a(n) = 10^n - n^10.

Original entry on oeis.org

1, 9, -924, -58049, -1038576, -9665625, -59466176, -272475249, -973741824, -2486784401, 0, 74062575399, 938082635776, 9862141508151, 99710745345024, 999423349609375, 9998900488372224, 99997984006099551, 999996429532773376

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, A024110. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[10^n-n^10: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011

lst={}; Do[AppendTo[lst,10^n-n^10],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[10^n-n^10,{n,0,20}] (* Harvey P. Dale, Apr 22 2018 *)

From Chai Wah Wu, Jan 26 2020: (Start)
a(n) = 21*a(n-1) - 165*a(n-2) + 715*a(n-3) - 1980*a(n-4) + 3762*a(n-5) - 5082*a(n-6) + 4950*a(n-7) - 3465*a(n-8) + 1705*a(n-9) - 561*a(n-10) + 111*a(n-11) - 10*a(n-12) for n > 11.
G.f.: (9*x^11 + 10140*x^10 + 477332*x^9 + 4504245*x^8 + 12648018*x^7 + 11793648*x^6 + 3241104*x^5 + 23538*x^4 - 37875*x^3 - 948*x^2 - 12*x + 1)/((x - 1)^11*(10*x - 1)). (End)

A024152 a(n) = 12^n - n^12.

Original entry on oeis.org

1, 11, -3952, -529713, -16756480, -243891793, -2173796352, -13805455393, -68289495040, -277269756129, -938082635776, -2395420006033, 0, 83695120256591, 1227224552173568, 15277275236695743, 184602783918325760

Offset: 0

N. J. A. Sloane

Conjecture: satisfies a linear recurrence having signature (25, -234, 1222, -4147, 9867, -17160, 22308, -21879, 16159, -8866, 3510, -949, 157, -12). - Harvey P. Dale, Jan 27 2019

The conjecture above is correct. From the general formula for {a(n)} we can see that the roots for the characteristic polynomial are one 12 and thirteen 1's, so the characteristic polynomial is (x - 12)*(x - 1)^13 = x^14 - 25*x^13 + 234*x^12 - ... + 12, with corresponding recurrence coefficients 25, -234, ..., -12. - Jianing Song, Jan 28 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (25,-234,1222,-4147,9867,-17160,22308,-21879,16159,-8866,3510,-949,157,-12).

Cf. A024012, A024026, A058794, A024040, A024054, A024068, A024082, A024096, A024110, A024124, A024138. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[12^n-n^12: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011

Table[12^n-n^12,{n,0,30}] (* Harvey P. Dale, Jan 27 2019 *)

A024138 a(n) = 11^n - n^11.

Original entry on oeis.org

1, 10, -1927, -175816, -4179663, -48667074, -361025495, -1957839572, -8375575711, -29023111918, -74062575399, 0, 2395420006033, 32730551749894, 375700268413577, 4168598413556276, 45932137677527745, 505412756602986138

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, A024110, A024124. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

[11^n-n^11: n in [0..20]]; // Vincenzo Librandi, Jun 30 2011

Table[11^n - n^11, {n, 0, 20}] (* Harvey P. Dale, Jul 31 2013 *)

From Chai Wah Wu, Jan 26 2020: (Start)
a(n) = 23*a(n-1) - 198*a(n-2) + 946*a(n-3) - 2915*a(n-4) + 6237*a(n-5) - 9636*a(n-6) + 10956*a(n-7) - 9207*a(n-8) + 5665*a(n-9) - 2486*a(n-10) + 738*a(n-11) - 133*a(n-12) + 11*a(n-13) for n > 12.
G.f.: (-12*x^12 - 22383*x^11 - 1677037*x^10 - 24085511*x^9 - 104916261*x^8 - 163227822*x^7 - 91395930*x^6 - 14499462*x^5 + 523986*x^4 + 130461*x^3 + 1959*x^2 + 13*x - 1)/((x - 1)^12*(11*x - 1)). (End)

A024017 a(n) = 2^n - n^7.

Original entry on oeis.org

1, 1, -124, -2179, -16368, -78093, -279872, -823415, -2096896, -4782457, -9998976, -19485123, -35827712, -62740325, -105397120, -170826607, -268369920, -410207601, -611957888, -893347451, -1278951424, -1798991389, -2490163584, -3396436839

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..239
Index entries for linear recurrences with constant coefficients, signature (10,-44,112,-182,196,-140,64,-17,2).

Cf. sequences of the form k^n-n^7: this sequence (k=2), A024030 (k=3), A024043 (k=4), A024056 (k=5), A024069 (k=6), A024082 (k=7), A024095 (k=8), A024108 (k=9), A024121 (k=10), A024134 (k=11), A024147 (k=12).

[2^n-n^7: n in [0..25]]; // Vincenzo Librandi, Apr 30 2011

I:=[1,1,-124,-2179,-16368,-78093,-279872,-823415,-2096896]; [n le 9 select I[n] else 10*Self(n-1)-44*Self(n-2)+112*Self(n-3)-182*Self(n-4)+196*Self(n-5)-140*Self(n-6)+64*Self(n-7)-17*Self(n-8)+2*Self(n-9): n in [1..35]]; // Vincenzo Librandi, Oct 07 2014

Table[2^n - n^7, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 9 x - 90 x^2 - 1007 x^3 + 36 x^4 + 3585 x^5 + 2290 x^6 + 231 x^7 + 3 x^8)/((1 - 2 x) (1 - x)^8), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 07 2014 *)
LinearRecurrence[{10,-44,112,-182,196,-140,64,-17,2},{1,1,-124,-2179,-16368,-78093,-279872,-823415,-2096896},30] (* Harvey P. Dale, Feb 28 2023 *)

G.f.: (1-9*x-90*x^2-1007*x^3+36*x^4+3585*x^5+ 2290*x^6 +231*x^7+3*x^8)/((1-2*x)*(1-x)^8). - Vincenzo Librandi, Oct 07 2014

a(n) = 10*a(n-1)-44*a(n-2)+112*a(n-3)-182*a(n-4)+196*a(n-5)-140*a(n-6)+64*a(n-7)-17*a(n-8)+2*a(n-9). - Vincenzo Librandi, Oct 07 2014

A337670 Numbers that can be expressed as both Sum x^y and Sum y^x where the x^y are not equal to y^x for any (x,y) pair and all (x,y) pairs are distinct.

Original entry on oeis.org

432, 592, 1017, 1040, 1150, 1358, 1388, 1418, 1422, 1464, 1554, 1612, 1632, 1713, 1763, 1873, 1889, 1966, 1968, 1973, 1990, 2091, 2114, 2190, 2291, 2320, 2364, 2451, 2589, 2591, 2612, 2689, 2697, 2719, 2753, 2775, 2803, 2813, 2883, 3087, 3127, 3141, 3146

Offset: 1

Matej Veselovac, Sep 15 2020

nonn

Numbers m of form m = Sum_{i=1...k} b_i^e_i = Sum_{i=1...k} e_i^b_i such that b_i^e_i != e_i^b_i, b_i > 1, e_i > 1, k = |{{b_i, e_i}, i = 1, 2, ...}|, k > 1.
Terms of the sequence relate to the Diophantine equation Sum_{i=1...k} x_i = 0, k > 1, x_i != 0, where x_i = (b_i^e_i - e_i^b_i) such that b_i > 1, e_i > 1 and (i != j) => ({b_i, e_i} != {b_j, e_j}). That is, we are observing linear combinations of elements from {(r^n - n^r) : n,r > 1} \ {0}, under given conditions.
For sums with k = 20 terms, one infinite family of examples is known: "2^(2t) + t^(4) + 2^(2t+8) + (t+4)^(4) + 2^(2t+16) + (t+8)^(4) + 2^(2t+32) + (t+16)^(4) + 2^(2t+34) + (t+17)^(4) + 4^(t+1) + (2t+2)^(2) + 4^(t+2) + (2t+4)^(2) + 4^(t+10) + (2t+20)^(2) + 4^(t+14) + (2t+28)^(2) + 4^(t+18) + (2t+36)^(2)" is a term of the sequence, for every t > 4.

			17 = 2^3 + 3^2 = 3^2 + 2^3 is not in the sequence because {2,3} = {3,2} are not distinct.
25 = 3^3 + 2^4 = 3^3 + 4^2 is not in the sequence because 3^3 = 3^3 and 2^4 = 4^2 are commutative.
The smallest term of the sequence is:
  a(1) = 432 = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7
             = 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2.
The smallest term that has more than one representation is:
  a(11) = 1554 = 3^2 + 7^2 + 6^3 + 2^8 + 4^5
               = 2^3 + 2^7 + 3^6 + 8^2 + 5^4,
  a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6
               = 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3.
Smallest terms with k = 5, 6, 7, 8, 9, 10 summands are:
  a(9)  = 1422 = 5^2 + 7^2 + 9^2 + 3^5 + 4^5
               = 2^5 + 2^7 + 2^9 + 5^3 + 5^4,
  a(1)  = 432  = 3^2 + 5^2 + 2^6 + 3^4 + 5^3 + 2^7
               = 2^3 + 2^5 + 6^2 + 4^3 + 3^5 + 7^2,
  a(2)  = 592  = 3^2 + 5^2 + 7^2 + 4^3 + 2^6 + 5^3 + 2^8
               = 2^3 + 2^5 + 2^7 + 3^4 + 6^2 + 3^5 + 8^2,
  a(11) = 1554 = 3^2 + 5^2 + 2^6 + 10^2 + 2^7 + 3^5 + 2^8 + 3^6
               = 2^3 + 2^5 + 6^2 + 2^10 + 7^2 + 5^3 + 8^2 + 6^3,
  a(14) = 1713 = 3^2 + 2^5 + 6^2 + 8^2 + 4^3 + 2^7 + 3^5 + 2^9 + 5^4
               = 2^3 + 5^2 + 2^6 + 2^8 + 3^4 + 7^2 + 5^3 + 9^2 + 4^5,
  a(28) = 2451 = 3^2 + 5^2 + 6^2 + 8^2 + 3^4 + 2^7 + 6^3 + 3^5 + 5^4 + 2^10
               = 2^3 + 2^5 + 2^6 + 2^8 + 4^3 + 7^2 + 3^6 + 5^3 + 4^5 + 10^2.

Math StackExchange, Base-Exponent Invariants, 2020.
Matej Veselovac, PYTHON program for A337670
Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A337671 (subsequence for k <= 5).
Cf. A005188 (perfect digital invariants).
Cf. Perfect powers: A001597, A072103.
Cf. Commutative powers: A271936.
Cf. Nonnegative numbers of the form (r^n - n^r), for n,r > 1: A045575.
Cf. Numbers of the form (r^n - n^r): A024012 (r = 2), A024026 (r = 3), A024040 (r = 4), A024054 (r = 5), A024068 (r = 6), A024082 (r = 7), A024096 (r = 8), A024110 (r = 9), A024124 (r = 10), A024138 (r = 11), A024152 (r = 12).

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

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

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A024124 a(n) = 10^n - n^10.

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A024152 a(n) = 12^n - n^12.

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A024138 a(n) = 11^n - n^11.

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A024017 a(n) = 2^n - n^7.

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A337670 Numbers that can be expressed as both Sum x^y and Sum y^x where the x^y are not equal to y^x for any (x,y) pair and all (x,y) pairs are distinct.

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