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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

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Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

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

Programs

Formula

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)

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

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Author

Matej Veselovac, Sep 15 2020

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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