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 1-4 of 4 results.

A098682 Smallest prime larger than n^n.

Original entry on oeis.org

2, 5, 29, 257, 3137, 46663, 823547, 16777259, 387420499, 10000000019, 285311670673, 8916100448291, 302875106592269, 11112006825558043, 437893890380859403, 18446744073709551629, 827240261886336764251, 39346408075296537575531, 1978419655660313589123997
Offset: 1

Views

Author

Olaf Voß, Oct 27 2004

Keywords

Links

Crossrefs

Cf. A000312 (n^n), A074966, A098681, A116481, A161503, A333184.

Programs

Formula

a(n) = A074966(n) + n^n. - Michel Marcus, Mar 11 2020

A098681 Largest prime smaller than n^n.

Original entry on oeis.org

3, 23, 251, 3121, 46649, 823541, 16777213, 387420479, 9999999967, 285311670569, 8916100448237, 302875106592241, 11112006825557999, 437893890380859323, 18446744073709551557, 827240261886336764159, 39346408075296537575359
Offset: 2

Views

Author

Olaf Voß, Oct 27 2004

Keywords

Links

Crossrefs

Cf. A000312, A074967, A098682, A161503, A333184.

Programs

  • Mathematica
    PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k];f[n_]:=n^n;lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,30}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)
Table[NextPrime[n^n,-1],{n,2,20}] (* Harvey P. Dale, Dec 02 2017 *)

A333185 Numbers k such that k^k is the average of its nearest 2 primes.

Original entry on oeis.org

2, 6, 9, 940
Offset: 1

Views

Author

Hugo Pfoertner, Mar 11 2020

Keywords

Examples

			          Previous P         k^k           Next P
  a(n)  A098681(a(n))  A000312(a(n))  A098682(a(n))
    2               3              4              5
    6           46649          46656          46663
    9       387420479      387420489      387420499
  940    940^940-3063        940^940   940^940+3063

Crossrefs

Cf. A000312, A074966, A074967, A098681, A098682, A161503, A333184.

Programs

  • PARI
    isok(k) = if (k>1, my(x=k^k); precprime(x-1)+nextprime(x+1) == 2*x); \\ Michel Marcus, Mar 14 2020

Formula

A333184(a(n)) = 0.
A074966(a(n)) = A074967(a(n)).

A333959 First occurrence of n in A334144.

Original entry on oeis.org

1, 6, 15, 33, 65, 77, 154, 161, 217, 231, 455, 469, 483, 693, 957, 987, 1001, 1449, 1463, 2021, 2717, 2093, 2415, 2967, 3003, 4147, 3059, 4853, 4945, 4899, 6083, 8533, 4991, 7161, 9982, 8987, 9177, 10787, 10857, 10465, 10199, 12857, 14539, 20355, 18753, 20398
Offset: 1

Views

Author

Michael De Vlieger, Peter Kagey, Antti Karttunen, Robert G. Wilson v, Apr 24 2020

Keywords

Comments

Consider the mappings f(m) := m -> m - m/p across primes p | m.
Row m of A334184, read as a triangle T(m, k), lists the number of distinct values that proceed from the mapping after exactly k iterations.
A334144(m) is the largest value in row m of A334184.
The smallest term in this sequence that is not an index of a record in A334144 is a(22) = 2093.
From Robert G. Wilson v, Jun 14 2020: (Start)
All terms are nonprimes, but not necessarily squarefree. They are: 693, 1449, 91791, 13126113, 46334057, ..., .
Even terms: 6, 154, 9982, 20398, 29946, 812630, 1366666, 4263182, 17766658, 22866158, 34688186, 80633294, ..., .
Except for the initial even term, all even terms divided by 2 are also terms.
(End)

Examples

			1 is the first term since 1 is the empty product.
6 follows 1 since 2 <= m <= 5 have total order, thus the maximum number in A333184 is 1. For m = 6, the mapping f(m) has two distinct results {4, 3}, which generate chains {4, 2, 1} and {3, 2, 1}, respectively, with the last two terms in both chains coincident. Since the largest number of terms in an antichain is 2, a(2) = 6.
15 follows 6 since row 15 of A334184 = [1, 2, 3, 2, 1, 1] is the smallest m for which n = 3 appears.
Hasse diagrams of the 3 smallest terms, with brackets around the widest row.
[1]        6           15
          / \          /\
         /   \        /  \
        [4   3]     12  __10
         |  /       | \/   |
         | /        |_/\   |
         2         [8  _6  5]
         |          | /_|_/
         |          |// |
         1          4   3
                    |  /
                    |_/
                    2
                    |
                    |
                    1

Links

Crossrefs

Cf. A064097, A332809, A333123, A334144, A334184.

Programs

  • Mathematica
    With[{s = Table[Max[Length@ Union@ # & /@ Transpose@ #] &@ If[n == 1, {{1}}, NestWhile[If[Length[#] == 0, Map[{n, #} &, # - # /FactorInteger[#][[All, 1]] ], Union[Join @@ Map[Function[{w, n}, Map[Append[w, If[n == 0, 0, n - n/#]] &, FactorInteger[n][[All, 1]] ]] @@ {#, Last@ #} &, #]] ] &, n, If[ListQ[#], AllTrue[#, Last[#] > 1 &], # > 1] &]], {n, 10^3}]}, TakeWhile[Array[FirstPosition[s, #][[1]] &, Max@ s], IntegerQ]]
f[n_] := Block[{lst = {{n}}}, While[lst[[-1]] != {1}, lst = Join[ lst, {Union[ Flatten[# - #/(First@# & /@ FactorInteger@#) & /@ lst[[-1]]] ]}]]; Max[Length@# & /@ lst]]; t[] := 0; k = 1; While[k < 21001, a = f@k; If[ t[a] == 0, t[a] = k]; k++]; t@# & /@ Range@ 46 (* _Robert G. Wilson v, Jun 14 2020 *)
Showing 1-4 of 4 results.

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