cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A122003 Numbers n such that A024152(n) = 12^n - n^12 is prime.

Original entry on oeis.org

1, 13, 25, 325, 833, 2087, 29773
Offset: 1

Views

Author

Alexander Adamchuk, Sep 12 2006

Keywords

Comments

Corresponding primes of the form A024152[n] = 12^n - n^12 are {11,83695120256591,953962166381085484825907807,...}.
a(8) > 50000. - Michael S. Branicky, Oct 01 2024

Crossrefs

Cf. A024152, A072180, A109387.

Programs

  • Mathematica
    Do[f=12^n-n^12;If[PrimeQ[f],Print[{n,f}]],{n,1,833}]
  • PARI
    is(n)=isprime(12^n-n^12) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(6) from Robert G. Wilson v, Sep 14 2006
a(7) from Donovan Johnson, Feb 26 2008

A122735 Smallest prime of the form (n^k - k^n) for k > 1, or 1 if no such prime exists.

Original entry on oeis.org

1, 7, 17, 1, 6102977801, 162287, 79792265017612001, 8375575711, 2486784401
Offset: 1

Views

Author

Alexander Adamchuk, Sep 24 2006, corrected Mar 03 2007

Keywords

Comments

a(10) = 10^273 - 273^10 is too large to include.
a(16) = 1 because primes of the form (16^k - k^16) do not exist, since 16^k - k^16 = (4^k - k^4)(4^k + k^4).
The corresponding numbers k such that a(n) = (n^k - k^n) are listed in A128355, where k = 0 corresponds to a(n) = 1.
Currently a(n) is not known for n = {17, 18, 22, 25, 26, 27, 28, ...}.

Examples

			a(1) = 1 because (1^k - k^1) = (1 - k) < 0 for k > 1.
a(2) = 7 because 2^5 - 5^2 = 7 is prime, but (2^k - k^2) is not prime for 1 < k < 5, (2^2 - 2^2) = 0, (2^3 - 3^2) = -1, (2^4 - 4^2) = 0.
a(4) = 1 because no prime of the form (4^k - k^4) exists; 4^k - k^4 = (2^k - k^2)*(2^k + k^2).
a(12) = 83695120256591 = 12^13 - 13^12 = A024152(A122003(2)).

Crossrefs

Cf. A024152, A122003.
Cf. A128355.

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

Views

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.

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