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.

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A274354 Number of factors L(i) > 1 of A274281(n), where L = A000032 (Lucas numbers, 2,1,3,4,..., with 1 excluded).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 3, 2, 1, 2, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 2, 2, 1, 2, 3, 2, 3, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 3, 3, 2, 3, 4, 3, 4, 2, 3, 2, 1, 2, 2, 3, 3, 2, 3, 3, 4, 3, 2, 3, 4, 3
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2016

Keywords

Examples

			The products of distinct Lucas numbers (including 2, excluding 1), arranged in increasing order, comprise A274281 (with 1 removed).  The list begins with 2, 3, 4, 6 = 2*3, 7, 8 = 2*4, 11, 12, 14, 18, 21, 22, 24 = 2*3*4, so that a(4) = 2, a(6) = 2, a(13) = 3.

Crossrefs

Cf. A000032, A274353, A274349, A274350, A160009, A272947.

Programs

  • Mathematica
    r[1] := 2; r[2] := 1; r[n_] := r[n] = r[n - 1] + r[n - 2];
s = {1}; z = 40; f = Join[{2}, Map[r, 2 + Range[z]]]; Take[f, 10]
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
infQ[n_] := MemberQ[f, n];
ans = Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[
Rest[Subsets[Map[#[[1]] &, Select[Map[{#, infQ[#]} &,
Divisors[s[[n]]]], #[[2]] && #[[1]] > 1 &]]]]], {n, 2, 200}];
Take[ans, 10]
w = Map[Length, ans]
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A274349 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A274350 *)
(* Peter J. C. Moses, Jun 17 2016 *)

A274371 Numbers that are a product of distinct Fibonacci numbers (A000045) and also a product of distinct Lucas numbers (A000032, including 2).

Original entry on oeis.org

1, 2, 3, 6, 8, 21, 24, 42, 126, 144, 168, 432, 504, 987, 1008, 1974, 3024, 5922, 7896, 23688, 46368, 47376, 139104, 142128, 973728, 2178309, 4356618, 13069854, 17426472, 45765216, 52279416, 104558832, 313676496, 4807526976, 14422580928, 100958066496
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2016

Keywords

Comments

Contains A273803 as a subsequence.

Examples

			504 = 3*8*21 = 4*7*18.

Crossrefs

Cf. A000032, A000045, A160009, A274280, A273803.

Programs

  • Mathematica
    s = {1}; z = 60; f = Fibonacci[2 + Range[z]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; s = Prepend[s, 0];  u = Take[s, 100]
g = LucasL[-1 + Range[z]]; t = {1}; Do[t = Union[t, Select[t*g[[i]], # <= g[[z]] &]], {i, z}]; w = Intersection[s, t]

A274453 Products of distinct numbers in A052963.

Original entry on oeis.org

2, 5, 10, 14, 28, 40, 70, 80, 115, 140, 200, 230, 331, 400, 560, 575, 662, 953, 1120, 1150, 1610, 1655, 1906, 2744, 2800, 3220, 3310, 4600, 4634, 4765, 5488, 5600, 7901, 8050, 9200, 9268, 9530, 13240, 13342, 13720, 15802, 16100, 22750, 23000, 23170, 26480
Offset: 1

Views

Author

Clark Kimberling, Jun 23 2016

Keywords

Examples

			The numbers in A274453 are 1, 2, 5, 14, 40, 115, 331,..., so that the sequence of all products of distinct members, in increasing order, is (2, 5, 10, 14, 28, 40, 70, 80,...).

Links

Crossrefs

Cf. A160009, A274280, A274432, A274452.

Programs

  • Mathematica
    r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = 3 r[n - 1] - r[n - 3]
s = {1}; z = 30; f = Map[r, Range[z]]; Take[f, 20] (* A052963 *)
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z]  (* A274453 *)

A274454 Products of distinct numbers in the Pell sequence (A000129).

Original entry on oeis.org

2, 5, 10, 12, 24, 29, 58, 60, 70, 120, 140, 145, 169, 290, 338, 348, 350, 408, 696, 700, 816, 840, 845, 985, 1680, 1690, 1740, 1970, 2028, 2030, 2040, 2378, 3480, 4056, 4060, 4080, 4200, 4756, 4896, 4901, 4925, 5741, 8400, 9792, 9802, 9850, 10140, 10150
Offset: 1

Views

Author

Clark Kimberling, Jun 23 2016

Keywords

Examples

			The numbers in A274454 are 1, 2, 5, 12, 29, 70, 169, 408,..., so that the sequence of all products of distinct members, in increasing order, is (2, 5, 10, 12, 24, 29, 58, 60,...).

Links

Crossrefs

Cf. A160009, A274280.

Programs

  • Mathematica
    r[1] = 1; r[2] = 2; r[n_] := r[n] = 2 r[n - 1] + r[n - 2]
s = {1}; z = 30; f = Map[r, Range[z]]; Take[f, 20] (* A000129 *)
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z]  (* A274454 *)

A309840 If n = Sum (2^e_k) then a(n) = Product (Fibonacci(e_k + 3)).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 8, 16, 24, 48, 40, 80, 120, 240, 13, 26, 39, 78, 65, 130, 195, 390, 104, 208, 312, 624, 520, 1040, 1560, 3120, 21, 42, 63, 126, 105, 210, 315, 630, 168, 336, 504, 1008, 840, 1680, 2520, 5040, 273, 546, 819, 1638, 1365, 2730, 4095, 8190
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 19 2019

Keywords

Examples

			23 = 2^0 + 2^1 + 2^2 + 2^4 so a(23) = Fibonacci(3) * Fibonacci(4) * Fibonacci(5) * Fibonacci(7) = 390.

Links

  • Michael De Vlieger, Table of n, a(n) for n = 0..16384
  • Michael De Vlieger, Fan style binary tree showing a(n) for n = 0..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, where magenta additionally represents powerful numbers that are not perfect prime powers and bright green represents primorials.

Crossrefs

Cf. A000045, A003266, A019565, A022290, A029930, A121663, A160009.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[Product[(1 + Fibonacci[k + 3] x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := Fibonacci[Floor[Log[2, n]] + 3] a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 55}]
  • PARI
    a(n)={vecprod([fibonacci(k+2) | k<-Vec(select(b->b, Vecrev(digits(n, 2)), 1))])} \\ Andrew Howroyd, Aug 19 2019

Formula

G.f.: Product_{k>=0} (1 + Fibonacci(k + 3) * x^(2^k)).
a(0) = 1; a(n) = Fibonacci(floor(log_2(n)) + 3) * a(n - 2^floor(log_2(n))).
a(2^(k-2)-1) = A003266(k).

A376807 Products of distinct prime Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 89, 130, 178, 195, 233, 267, 390, 445, 466, 534, 699, 890, 1157, 1165, 1335, 1398, 1597, 2314, 2330, 2670, 3029, 3194, 3471, 3495, 4791, 5785, 6058, 6942, 6990, 7985, 9087, 9582, 11570, 15145, 15970, 17355, 18174
Offset: 1

Views

Author

Jack Brennen, Oct 04 2024

Keywords

Comments

Each term is a product of a finite subsequence of A005478.

Crossrefs

Cf. A005478, A160009.

Programs

  • Python
    import itertools, math, sympy
def fibprimegen(limit):  # Generate Fibonacci primes <= limit
  a,b = 1,2
  while b <= limit:
    if sympy.isprime(b):
      yield b
    a,b = b,a+b
LIMIT=1000000
fibprimes=list(fibprimegen(LIMIT))
fibprimeseqs=itertools.chain.from_iterable(
    itertools.combinations(fibprimes,n) for n in range(len(fibprimes)+1))
print(sorted(a for a in map(math.prod,fibprimeseqs) if a <= LIMIT))
Previous Showing 21-26 of 26 results.

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