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.

A330999 Infinitary Ruth-Aaron numbers: numbers k such that A181894(k) = A181894(k+1).

Original entry on oeis.org

5, 77, 714, 948, 2431, 2491, 2996, 3450, 4293, 5405, 5560, 5885, 5959, 11124, 13869, 14587, 16932, 17080, 17346, 18468, 19551, 26642, 31931, 33019, 37925, 42250, 47544, 48635, 49240, 52554, 53192, 60048, 79248, 80837, 89979, 95709, 98119, 98644, 99163, 108458
Offset: 1

Views

Author

Amiram Eldar, Jan 05 2020

Keywords

Comments

A variation of Ruth-Aaron numbers with "Fermi-Dirac primes" (or infinitary components) instead of prime divisors.

Examples

			5 is a term since A181894(5) = A181894(6) = 5.

Links

Crossrefs

Cf. A006145, A039752, A050376, A181894, A331000.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); s[1] = 0; s[n] := Plus @@ (Flatten @ (f @@@ FactorInteger[n])); seq ={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq

A333802 Numbers k such that A181894(k)+1 = A181894(k+1).

Original entry on oeis.org

2, 3, 4, 16, 20, 35, 143, 152, 208, 256, 650, 1624, 2232, 4233, 4345, 5368, 8099, 9424, 11024, 11919, 12099, 14905, 18424, 20220, 21716, 22194, 24335, 25592, 26123, 27390, 30457, 34945, 38180, 40425, 51992, 52206, 52947, 56563, 63712, 65536, 67123, 71154, 71284
Offset: 1

Views

Author

Amiram Eldar, Apr 05 2020

Keywords

Comments

A variation of A064111 and A228126 with "Fermi-Dirac primes" (or infinitary components) instead of prime divisors.

Examples

			4 is a term since A181894(4) + 1 = 4 + 1 = 5 = A181894(5).

Links

Crossrefs

Cf. A006145, A039752, A050376, A064111, A181894, A228126, A330999, A333801.

Programs

  • Mathematica
    f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); s[1] = 0; s[n_] := Plus @@ (Flatten @ (f @@@ FactorInteger[n])); seq = {}; s1 = 0; Do[s2 = s[n]; If[s1 + 1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10^5}]; seq

A213925 Triangle read by rows: n-th row contains Fermi-Dirac representation of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 3, 7, 2, 4, 9, 2, 5, 11, 3, 4, 13, 2, 7, 3, 5, 16, 17, 2, 9, 19, 4, 5, 3, 7, 2, 11, 23, 2, 3, 4, 25, 2, 13, 3, 9, 4, 7, 29, 2, 3, 5, 31, 2, 16, 3, 11, 2, 17, 5, 7, 4, 9, 37, 2, 19, 3, 13, 2, 4, 5, 41, 2, 3, 7, 43, 4, 11, 5, 9, 2, 23, 47, 3, 16, 49, 2, 25
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 20 2013

Keywords

Comments

Unique factorization of n into distinct prime powers of form p^(2^k), cf. A050376.

Examples

			First rows:
.     1:    1
.     2:    2
.     3:    3
.     4:    4
.     5:    5
.     6:    2  3
.     7:    7
.     8:    2  4                   8 = 2^2^0 * 2^2^1
.     9:    9
.    10:    2  5
.......
.   990:    2   5  9  11
.   991:  991
.   992:    2  16 31             992 = 2^2^0 * 2^2^2 * 31^2^0
.   993:    3 331
.   994:    2   7 71
.   995:    5 199
.   996:    3   4 83
.   997:  997
.   998:    2 499
.   999:    3   9 37             999 = 3^2^0 * 3^2^1 * 37^2^0
.  1000:    2   4  5  25        1000 = 2^2^0 * 2^2^1 * 5^2^0 * 5^2^1 .

Links

Crossrefs

Cf. A050376.
For n > 1: A064547 (row lengths), A181894 (row sums), A223490, A223491.

Programs

  • Haskell
    a213925 n k = a213925_row n !! (k-1)
a213925_row 1 = [1]
a213925_row n = reverse $ fd n (reverse $ takeWhile (<= n) a050376_list)
   where fd 1 _      = []
         fd x (q:qs) = if m == 0 then q : fd x' qs else fd x qs
                       where (x',m) = divMod x q
a213925_tabf = map a213925_row [1..]
  • Maple
    T:= n-> `if`(n=1, [1], sort([seq((l-> seq(`if`(l[j]=1, i[1]^(2^(j-1)), [][]),
             j=1..nops(l)))(convert(i[2], base, 2)), i=ifactors(n)[2])]))[]:
seq(T(n), n=1..60);  # Alois P. Heinz, Feb 20 2018
  • Mathematica
    nmax = 50; FDPrimes = Reap[k = 1; While[lim = nmax^(1/k); lim > 2, Sow[Prime[Range[PrimePi[lim]]]^k]; k = 2 k]][[2, 1]] // Flatten // Union;
f[1] = 1; f[n_] := Reap[m = n; Do[If[m == 1, Break[], If[Divisible[m, p], m = m/p; Sow[p]]], {p, Reverse[FDPrimes]}]][[2, 1]] // Reverse;
Array[f, nmax] // Flatten (* Jean-François Alcover, Feb 05 2019 *)
  • PARI
    row(n) = if(n == 1, [1], my(f = factor(n), p = f[, 1], e = f[, 2], r = [], b); for(i = 1, #p, b = binary(e[i]); for(j = 0, #b-1, if(b[#b-j], r = concat(r, p[i]^(2^j))))); r); \\ Amiram Eldar, May 02 2025

Formula

Product_{k=1..A064547(n)} T(n,k) = n.

Extensions

Example corrected (row 992) by Reinhard Zumkeller, Mar 11 2015
Showing 1-3 of 3 results.

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