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 1-10 of 10 results.

A274353 Number of factors L(i) > 1 of A274280(n), where L = A000032 (Lucas numbers, 1,3,4,...)

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jun 18 2016

Keywords

Examples

			The products of distinct Lucas numbers, arranged in increasing order, comprise A274280.  The list begins with 1, 3, 4, 7, 11, 12 = 3*4, so that a(6) = 2.

Links

Crossrefs

Cf. A000032, A274280, A274347, A274348, A160009, A272947, A274354.

Programs

  • Mathematica
    r[1] := 1; r[2] := 3; r[n_] := r[n] = r[n - 1] + r[n - 2];
s = {1}; z = 40; f = Map[r, Range[z]]; Take[f, 10]
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
infQ[n_] := MemberQ[f, n];
a = 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}];
ans = Join[{{1}}, a]; Take[ans, 8]
w = Map[Length, ans] (* A274353 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A274347 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A274348 *)
(* Peter J. C. Moses, Jun 17 2016 *)

A273803 Numbers that are a product of distinct Fibonacci numbers (A160009) and also a product of distinct Lucas numbers (A274280).

Original entry on oeis.org

1, 3, 21, 126, 504, 987, 5922, 23688, 2178309, 13069854, 52279416, 10610209857723, 63661259146338, 254645036585352
Offset: 1

Views

Author

Clark Kimberling, Jun 19 2016

Keywords

Comments

Is every term greater than 3 divisible by 21?

Examples

			126 = 2*3*21 = 7*18.

Crossrefs

Cf. A000204, A000045, A160009, A274280, A274371.

Programs

  • Mathematica
    s = {1}; z = 70; f = Fibonacci[2 + Range[z]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; s = Prepend[s, 0];  (* A160009 *)
g = LucasL[Range[z]]; t = {1}; Do[t = Union[t, Select[t*g[[i]], # <= g[[z]] &]], {i, z}];
Intersection[s, t]

A160009 Numbers that are the product of distinct Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 13, 15, 16, 21, 24, 26, 30, 34, 39, 40, 42, 48, 55, 63, 65, 68, 78, 80, 89, 102, 104, 105, 110, 120, 126, 130, 144, 165, 168, 170, 178, 195, 204, 208, 210, 233, 240, 267, 272, 273, 275, 288, 312, 315, 330, 336, 340, 377, 390, 432, 440, 442, 445
Offset: 1

Views

Author

T. D. Noe, Apr 29 2009

Keywords

Comments

Starts the same as A049862, the product of two distinct Fibonacci numbers. This sequence has an infinite number of consecutive terms that are consecutive numbers (such as 15 and 16) because fib(k)*fib(k+3) and fib(k+1)*fib(k+2) differ by one for all k >= 0.
It follows from Carmichael's theorem that if u and v are finite sets of Fibonacci numbers such that (product of all the numbers in u) = (product of all the numbers in v), then u = v. The same holds for many other 2nd order linear recurrence sequences with constant coefficients. In the following guide to related "distinct product sequences", W = Wythoff array, A035513:
base sequence distinct-product sequence
A000045 (Fibonacci) A160009
A000032 (Lucas, without 2) A274280
A000032 (Lucas, with 2) A274281
A000285 (1,4,5,...) A274282
A022095 (1,5,6,...) A274283
A006355 (2,4,6,...) A274284
A013655 (2,5,7,...) A274285
A022086 (3,6,9,...) A274191
row 2 of W: (4,7,11,...) A274286
row 3 of W: (6,10,16,...) A274287
row 4 of W: (9,15,24,...) A274288
- Clark Kimberling, Jun 17 2016

Links

Crossrefs

A059844, A065108

Programs

  • Mathematica
    s={1}; nn=30; f=Fibonacci[2+Range[nn]]; Do[s=Union[s,Select[s*f[[i]],#<=f[[nn]]&]], {i,nn}]; s=Prepend[s,0]

A274432 Products of distinct tribonacci numbers (A000213).

Original entry on oeis.org

3, 5, 9, 15, 17, 27, 31, 45, 51, 57, 85, 93, 105, 135, 153, 155, 171, 193, 255, 279, 285, 315, 355, 459, 465, 513, 525, 527, 579, 653, 765, 837, 855, 945, 965, 969, 1065, 1201, 1395, 1539, 1575, 1581, 1737, 1767, 1775, 1785, 1959, 2209, 2295, 2565, 2635
Offset: 1

Views

Author

Clark Kimberling, Jun 22 2016

Keywords

Examples

			The tribonacci numbers are 1,1,1,3,5,9,17,31,..., so that the sequence of all products of distinct members, in increasing order, is (3, 5, 9, 15, 17, 27, 31, 45,...).

Links

Crossrefs

Cf. A160009, A274280, A274433 (binary products), A274434 (trinary products).

Programs

  • Mathematica
    r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = r[n - 1] + r[n - 2] + r[n - 3];
s = {1}; z = 60; f = Map[r, Range[z]]; Take[f, 20]
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z]  (*A274432*)
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, 300}];
Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A274433 *)
Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A274434 *)
(* Peter J. C. Moses, Jun 17 2016 *)

A274281 Numbers that are a product of distinct Lucas numbers (2,1,3,4,7,11,...)

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 11, 12, 14, 18, 21, 22, 24, 28, 29, 33, 36, 42, 44, 47, 54, 56, 58, 66, 72, 76, 77, 84, 87, 88, 94, 108, 116, 123, 126, 132, 141, 144, 152, 154, 168, 174, 188, 198, 199, 203, 216, 228, 231, 232, 246, 252, 264, 282, 304, 308, 319, 322
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2016

Keywords

Comments

See the Comment on distinct-product sequences in A160009.

Examples

			The Lucas numbers are 2,1,3,4,7,11,18,29,..., so that the sequence of all products of distinct Lucas numbers, in increasing order, are 1, 2, 3, 4, 6, 7, 8, 11, 12, 14, 18, 21, 22, 24, 28, 29,...

Links

Crossrefs

Cf. A160009, A274280.

Programs

  • Mathematica
    f[1] = 2; f[2] = 1; z = 32; f[n_] := f[n - 1] + f[n - 2]; f = Table[f[n], {n, 1, z}]; f
s = {1}; Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; s

A274286 Numbers that are a product of distinct numbers in row 2 of the Wythoff array, A035513.

Original entry on oeis.org

4, 7, 11, 18, 28, 29, 44, 47, 72, 76, 77, 116, 123, 126, 188, 198, 199, 203, 304, 308, 319, 322, 329, 492, 504, 517, 521, 522, 532, 792, 796, 812, 836, 843, 846, 861, 1276, 1288, 1316, 1353, 1363, 1364, 1368, 1386, 1393, 2068, 2084, 2088, 2128, 2189, 2204
Offset: 1

Views

Author

Clark Kimberling, Jun 17 2016

Keywords

Comments

See the Comment on distinct-product sequences in A160009.

Examples

			28 = 4*7, 308 = 4*7*11.

Links

Crossrefs

Cf. A160009, A274280.

Programs

  • Mathematica
    f[1] = 4; f[2] = 7; z = 33; f[n_] := f[n - 1] + f[n - 2]; f = Table[f[n], {n, 1, z}]; f
s = {1}; Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; s1 = Rest[s]

A274452 Products of distinct Narayana's cow numbers (A000930).

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 12, 13, 18, 19, 24, 26, 27, 28, 36, 38, 39, 41, 48, 52, 54, 56, 57, 60, 72, 76, 78, 82, 84, 88, 104, 108, 112, 114, 117, 120, 123, 129, 144, 152, 156, 162, 164, 168, 171, 176, 180, 189, 216, 224, 228, 234, 240, 246, 247, 252, 258, 264, 277
Offset: 1

Views

Author

Clark Kimberling, Jun 23 2016

Keywords

Examples

			The Narayana's cow numbers numbers are 1, 2, 3, 4, 6, 9, 13, 19, 28, ..., so that the sequence of all products of distinct members, in increasing order, is (2, 3, 4, 6, 8, 9, 12, 13, 18, 19, 24, ...).

Links

Crossrefs

Cf. A160009, A274280, A274432.

Programs

  • Mathematica
    r[1] := 1; r[2] := 1; r[3] = 1; r[n_] := r[n] = r[n - 1] + r[n - 3]
s = {1}; z = 60; f = Map[r, Range[z]]; Take[f, 20] (*A000930*)
Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}];
Take[s, 2 z] (* A274452 *)

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 *)
Showing 1-10 of 10 results.

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