A024026 -id:A024026 - cp's OEIS frontend

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

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)

A062278 a(n) = floor(3^n / n^3).

Original entry on oeis.org

3, 1, 1, 1, 1, 3, 6, 12, 27, 59, 133, 307, 725, 1743, 4251, 10509, 26285, 66430, 169450, 435848, 1129505, 2947131, 7737583, 20430377, 54226471, 144621405, 387420489, 1042127936, 2813988985, 7625597484, 20733556989, 56549688380

Offset: 1

Henry Bottomley, Jul 02 2001

nonn

3 is the only integer value of k for which floor(n^k / k^n) is always positive. For positive real x and k, the only value of k for which x^k is always greater than or equal to k^x is e = 2.71828...

			a(2) = floor(3^2 / 2^3) = floor(9/8) = 1.

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A000244, A000578, A024026, A060505.

List([1..35],n->Int(3^n/n^3)); # Muniru A Asiru, Jul 01 2018

seq(floor(3^n/n^3),n=1..35); # Muniru A Asiru, Jul 01 2018

Table[Floor[3^n/n^3],{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)

{ default(realprecision, 50); for (n=1, 200, write("b062278.txt", n, " ", floor(3^n / n^3)) ) } \\ Harry J. Smith, Aug 03 2009

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).

A168299 a(n) = 1 + 3^n * n^3.

Original entry on oeis.org

1, 4, 73, 730, 5185, 30376, 157465, 750142, 3359233, 14348908, 59049001, 235782658, 918330049, 3502727632, 13124466937, 48427561126, 176319369217, 634465620820, 2259436291849, 7971951402154, 27894275208001, 96873331012984

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 22 2009

Numerator of 3^(-n) + n^3.

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

Cf. A024026.

[1+n^3*3^n: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

f[n_]:=3^n-n^3; Table[Numerator[f[n]],{n,0,-50,-1}]

for(n=0,100,print1(1+(n^3)*(3^n),",")) \\ Edward Jiang, Nov 22 2013

From R. J. Mathar, Nov 24 2009: (Start)
a(n) = 1 + A062074(n).
G.f.: (1 -9*x +87*x^2 -117*x^3 +54*x^4)/((1-x)*(3*x-1)^4). (End)
E.g.f.: exp(x) + 3*x*(1 + 9*x + 9*x^2)*exp(3*x). - G. C. Greubel, Jul 17 2016

Corrected offset, and simplified the definition - R. J. Mathar, Nov 24 2009

cp's OEIS Frontend

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

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A062278 a(n) = floor(3^n / n^3).

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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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A168299 a(n) = 1 + 3^n * n^3.

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Magma

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