A303217 -id:A303217 - cp's OEIS frontend

A303215 A(n,k) is the n-th index of a Fibonacci number with exactly k prime factors (counted with multiplicity); square array A(n,k), n>=1, k>=1, read by antidiagonals.

3, 8, 4, 6, 9, 5, 20, 15, 10, 7, 18, 27, 16, 14, 11, 12, 44, 28, 21, 19, 13, 30, 40, 45, 32, 25, 22, 17, 54, 42, 50, 57, 52, 33, 26, 23, 24, 78, 56, 64, 63, 55, 35, 31, 29, 36, 80, 102, 66, 75, 68, 74, 37, 34, 43, 138, 100, 88, 128, 70, 92, 69, 77, 38, 41, 47

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   3,  8,  6, 20, 18,  12,  30,  54,  24,  36, ...
   4,  9, 15, 27, 44,  40,  42,  78,  80, 100, ...
   5, 10, 16, 28, 45,  50,  56, 102,  88, 114, ...
   7, 14, 21, 32, 57,  64,  66, 128, 110, 165, ...
  11, 19, 25, 52, 63,  75,  70, 130, 112, 174, ...
  13, 22, 33, 55, 68,  92,  81, 135, 184, 256, ...
  17, 26, 35, 74, 69,  95, 104, 147, 186, 266, ...
  23, 31, 37, 77, 76,  99, 105, 154, 189, 273, ...
  29, 34, 38, 85, 91, 116, 136, 170, 196, 282, ...
  43, 41, 39, 87, 98, 117, 148, 171, 225, 296, ...

Alois P. Heinz, Antidiagonals n = 1..16, flattened

Columns k=1-13 give: A001605, A072381, A114812, A114813, A114814, A114815, A114816, A114817, A114818, A114819, A114820, A114821, A114822.
Row n=1 gives A072396.
Cf. A000045, A001222, A038575, A303216, A303217.

F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))

A[n_, k_] := Module[{h, p, q = 2}, p[k] = {}; While[Length[p[k]]Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

A000045(A(n,k)) = A303216(n,k).

A001222(A000045(A(n,k))) = k.

A114823 Indices of Fibonacci numbers with 13 distinct prime factors.

Original entry on oeis.org

120, 200, 220, 228, 260, 368, 392, 405, 414, 434, 472, 492, 512, 536, 584, 585, 595, 610, 615, 618, 645, 654, 693, 741, 762, 777, 830, 867, 894, 904, 931, 942, 957, 962, 978, 1045, 1066, 1070, 1074, 1102, 1106, 1108, 1147, 1194, 1209, 1266, 1268, 1309, 1310, 1317

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

From Robert Israel, Aug 18 2015: (Start)
Numbers n such that A022307(n) = 13.
If n is in the sequence, then k*n is not in the sequence for k > 1.
This is because A000045(n) divides A000045(k*n) while Carmichael's theorem says A000045(k*n) has at least one primitive prime factor. (End)

			a(1)=120 because the 120th Fibonacci number consists of 13 distinct prime factors (i.e., 5358359254990966640871840 = 2^5 * 3^2 * 5 * 7 * 11 * 23 * 31 * 41 * 61 * 241 * 2161 * 2521 * 20641).

Amiram Eldar, Table of n, a(n) for n = 1..54
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A000045, A001221, A022307.

Column k=13 of A303217.

[n: n in [1..3*10^2] |(#(PrimeDivisors(Fibonacci(n)))) eq 13]; // Vincenzo Librandi, Aug 05 2015

select(t -> nops(numtheory:-factorset(combinat:-fibonacci(t)))=13, [$1..1000]); # Robert Israel, Aug 10 2015

Select[Range[1250], PrimeNu[Fibonacci[#]]==13&] (* Harvey P. Dale, Apr 30 2015 *)

n=1;while(n<265,if(omega(fibonacci(n))==13,print1(n,", "));n++)

for n in range(1,3*10^2):
    if len(prime_factors(fibonacci(n)))==13:
        print(n) # Manfred Scheucher, Aug 04 2015

More terms from Ryan Propper, Apr 26 2006
a(36)-a(45) from Max Alekseyev, Aug 18 2013
a(46)-a(50) from Amiram Eldar, Oct 14 2019

A114836 Indices of Fibonacci numbers with 9 distinct prime factors.

Original entry on oeis.org

80, 84, 88, 96, 100, 108, 132, 138, 156, 184, 189, 196, 222, 232, 243, 248, 250, 255, 268, 272, 273, 284, 286, 295, 318, 325, 328, 333, 357, 370, 402, 406, 412, 418, 426, 435, 477, 498, 534, 539, 556, 559, 561, 591, 602, 603, 604, 628, 629, 637, 639, 678

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=80 because the 80th Fibonacci number (i.e., 23416728348467685) consists of 9 distinct prime factors (i.e., 3 x 5 x 7 x 11 x 41 x 47 x 1601 x 2161 x 3041).

Amiram Eldar, Table of n, a(n) for n = 1..94
Blair Kelly, Fibonacci and Lucas Factorizations.

Column k=9 of A303217.

n=1;while(n<335,if(omega(fibonacci(n))==9,print1(n,", "));n++)

More terms from Ryan Propper, Apr 26 2006

A114841 Indices of Fibonacci numbers with 3 distinct prime factors.

Original entry on oeis.org

15, 16, 18, 21, 33, 35, 37, 38, 39, 46, 49, 51, 58, 62, 65, 67, 82, 86, 103, 106, 119, 122, 125, 139, 142, 145, 158, 166, 179, 181, 226, 233, 235, 241, 257, 263, 274, 281, 299, 301, 317, 337, 383, 389, 419, 457, 463, 473, 479, 491, 521, 541, 557, 619, 643, 659, 706, 719, 739, 751, 857, 863, 877, 881, 883, 911, 947, 983, 1021, 1033, 1061, 1069, 1109, 1117, 1123, 1181, 1187, 1193, 1213, 1226

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1) = 15 because 15th Fibonacci number has 3 distinct prime factors (i.e., 610 = 2 * 5 * 61).

Amiram Eldar, Table of n, a(n) for n = 1..83
Blair Kelly, Fibonacci and Lucas Factorizations.
Prapanpong Pongsriiam, Fibonacci and Lucas Numbers which have Exactly Three Prime Factors and Some Unique Properties of F18 and L18, Fibonacci Quart. 57 (2019), no. 5, 130-144.

Cf. A114823, A114824, A114825, A114826, A114836, A114837, A114838, A114839, A114840.

Column k=3 of A303217.

[n: n in [1..350] |(#(PrimeDivisors(Fibonacci(n)))) eq 3]; // Vincenzo Librandi, Mar 26 2018

with(numtheory): with(combinat):
a:=n->`if`(nops(factorset(fibonacci(n)))=3,n,NULL); [seq(a(n),n=1..300)]; # Muniru A Asiru, Mar 25 2018

Select[Range[500], PrimeNu[Fibonacci[#]]==3 &] (* Vincenzo Librandi, Mar 26 2018 *)

n=1;while(n<340,if(omega(fibonacci(n))==3,print1(n,", "));n++)

More terms from Ryan Propper, Apr 26 2006

a(57)-a(80) from Max Alekseyev, Aug 18 2013

A114826 Indices of Fibonacci numbers with 10 distinct prime factors.

Original entry on oeis.org

90, 126, 165, 256, 258, 266, 296, 304, 322, 369, 375, 376, 424, 428, 429, 474, 484, 494, 506, 574, 578, 589, 633, 646, 652, 658, 663, 664, 668, 669, 670, 687, 703, 705, 716, 724, 742, 754, 779, 782, 807, 845, 875, 892, 906, 909, 921, 946, 956, 964, 986, 1004, 1054, 1075, 1084, 1119, 1126, 1132, 1149, 1169, 1171, 1214, 1222, 1227

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=90 because the 90th Fibonacci number (i.e., 2880067194370816120) consists of 10 distinct prime factors (i.e., 2^3 * 5 * 11 * 17 * 19 * 31 * 61 * 181 * 541 * 109441).

Amiram Eldar, Table of n, a(n) for n = 1..69
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=10 of A303217.

n=1;while(n<305,if(omega(fibonacci(n))==10,print1(n,", "));n++)

More terms from Ryan Propper, Apr 26 2006

a(52)-a(64) from Max Alekseyev, Aug 18 2013

A060320 Index of smallest Fibonacci number with exactly n distinct prime factors.

Original entry on oeis.org

1, 3, 8, 15, 20, 30, 40, 70, 60, 80, 90, 140, 176, 120, 168, 180, 324, 252, 240, 378, 450, 432, 552, 360, 420, 690, 504, 880, 630, 600, 756, 720, 900, 792, 840, 1296, 1050, 1350, 1140, 1080, 1200, 1824, 1260, 1512, 1320, 1560, 1680

Offset: 0

Labos Elemer, Mar 28 2001

From Jon E. Schoenfield, Dec 28 2016: (Start)
 Note that the presence of incompletely factored Fibonacci numbers with indices as low as 1301 does not prevent the drawing of conclusions such as "a(44) = 1320" with certainly. Using F(1301) as an example, the compact table of Fibonacci results at the Kelly site indicates that F(1301) = p*q*r*c where p=6400921, q=14225131397, r=100794731109596201, and c is a 238-digit unfactored composite number. The complete factorization of every Fibonacci number up to F(1000) is explicitly given elsewhere on the site, and those results allow quick verification that a(n) <= 900 for all n in [0..34], so 1301 cannot be a term unless F(1301) has at least 35 distinct prime factors, which would require c to have at least 32 distinct prime factors, at least one of which would have to be less than ceiling(c^(1/32)) = 26570323, but trial division of c by every prime less than 26570323 shows that c has no prime factors that small. Thus, while A022307(1301) is unknown, it is certain that 1301 is not a term in this sequence. Similarly, making use of known factors, it can be proved that F(n) cannot have 44 or more distinct prime factors for any n < 1320, so since F(1320) has exactly 44 distinct prime factors, it is established that a(44) = 1320. (End)
a(47) >= 2835, a(48..68) = (2040, 1800, 2736, 2730, 1890, 1980, 2520, 2280, 2100, 2160, 2640, 3300, 3060, 3150, 2520, 3120, 3696, 3240, 3990, 3360, 3420), a(69) >= 4400, a(75) = 4320, a(77) = 4200, a(79) = 3780. - Max Alekseyev, Feb 03 2025

			n=9: F(80) = 23416728348467685 = 3 * 5 * 7 * 11 * 41 * 47 * 1601 * 2161 * 3041.
n=25: F(690) = 2^3 * 5 * 11 * 31 * 61 * 137 * 139 * 461 * 691 * 829 * 1151 * 1381 * 4831 * 5981 * 18077 * 28657 * 186301 * 324301 * 686551 * 1485571 * 4641631 * 117169733521 * 2441738887963981 * 3490125311294161 * 25013864044961447973152814604981 is the smallest Fibonacci number with exactly 25 distinct prime factors.

Ron Knott, Fibonacci numbers with tables of F(0)-F(500)
Hisanori Mishima, Fibonacci numbers (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 480).

Cf. A001605, A005478, A022307, A051694, A060319.

Row n=1 of A303217.

First /@ SortBy[#, Last] &@ Map[First@ # &, Values@ GroupBy[#, Last]] &@ Table[{n - Boole[n == 2], #, PrimeNu@ #} &@ Fibonacci@ n, {n, 2, 300}] (* Michael De Vlieger, Feb 18 2017, Version 10 *)
Module[{ff=Table[{n,PrimeNu[Fibonacci[n]]},{n,1400}]},Table[ SelectFirst[ ff,#[[2]]==k&],{k,0,40}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 28 2018 *)

my(o=[],s); print1(1); for(n=1,20, s=0; until( o[s]==n, #o

a(n) = min (k : A022307(k) = n).

Corrected by Shyam Sunder Gupta, Jul 20 2002
Edited by M. F. Hasler, Nov 01 2012
a(35)-a(40), a(42), a(44) computed based on Kelly's data in A022307 by Jon E. Schoenfield, Dec 28 2016
a(41), a(43), a(45)-a(46) from Max Alekseyev, Feb 03 2025

A303218 A(n,k) is the n-th Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

2, 21, 3, 610, 34, 5, 6765, 987, 55, 8, 832040, 46368, 2584, 144, 13, 102334155, 14930352, 196418, 10946, 377, 89, 190392490709135, 4807526976, 267914296, 317811, 3524578, 4181, 233, 1548008755920, 37889062373143906, 86267571272, 701408733, 2178309, 9227465, 17711, 1597

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   2,   21,     610,        6765,      832040,        102334155, ...
   3,   34,     987,       46368,    14930352,       4807526976, ...
   5,   55,    2584,      196418,   267914296,      86267571272, ...
   8,  144,   10946,      317811,   701408733,     225851433717, ...
  13,  377, 3524578,     2178309,  1134903170,   10610209857723, ...
  89, 4181, 9227465, 32951280099, 12586269025, 8944394323791464, ...

Alois P. Heinz, Antidiagonals n = 1..18, flattened

Column k=3 gives A137563.
Row n=1 gives: A060319.
Cf. A000045, A001221, A303216, A303217.

F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))

nmax = 12(*rows*);
maxIndex = 200; (* increase if message "part does not exist" *)
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k &];
T = Array[col, nmax];
A[n_, k_] := Fibonacci[T[[k, n]]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Feb 05 2021 *)

A(n,k) = A000045(A303217(n,k)).

A001221(A(n,k)) = k.

A114824 Indices of Fibonacci numbers with 12 distinct prime factors.

Original entry on oeis.org

176, 192, 198, 224, 297, 315, 364, 385, 410, 430, 441, 455, 496, 555, 582, 606, 609, 621, 627, 638, 682, 710, 711, 726, 729, 734, 759, 775, 805, 814, 826, 846, 854, 897, 903, 908, 932, 935, 981, 987, 994, 1001, 1002, 1022, 1025, 1067, 1086, 1089, 1105, 1244, 1246

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

			a(1)=176 because 176th Fibonacci number (i.e. 2706074082469569338358691163510069157) consists of 12 distinct prime factors (i.e. 3 * 7 * 43 * 47 * 89 * 199 * 263 * 307 * 881 * 967 * 93058241 * 56241856)

Amiram Eldar, Table of n, a(n) for n = 1..56
Blair Kelly, Fibonacci and Lucas Factorizations.

Column k=12 of A303217.

Flatten[Position[Fibonacci[Range[1110]],?(PrimeNu[#]==12&)]] (* _Harvey P. Dale, Jan 17 2015 *)

n=1;while(n<320,if(omega(fibonacci(n))==12,print1(n,", "));n++)

More terms from Ryan Propper, Apr 26 2006
a(42)-a(49) from Max Alekseyev, Aug 17 2013
a(50)-a(51) from Amiram Eldar, Oct 14 2019

A114825 Indices of Fibonacci numbers with 11 distinct prime factors.

Original entry on oeis.org

140, 144, 150, 160, 162, 190, 204, 208, 230, 234, 261, 285, 306, 310, 345, 351, 354, 372, 387, 388, 399, 438, 459, 469, 470, 475, 483, 485, 488, 507, 508, 518, 531, 549, 568, 596, 598, 632, 656, 671, 686, 688, 715, 788, 806, 808, 822, 834, 844, 872, 873, 874, 902, 925, 1005, 1048, 1052, 1058, 1073, 1076, 1083, 1101, 1113, 1124, 1131, 1146, 1158, 1162, 1166, 1173, 1195, 1205, 1221

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=140 because the 140th Fibonacci number (i.e., 81055900096023504197206408605) consists of 11 distinct prime factors (i.e., 3 * 5 * 11 * 13 * 29 * 41 * 71 * 281 * 911 * 141961 * 12317523121).

Amiram Eldar, Table of n, a(n) for n = 1..81
Blair Kelly, Fibonacci and Lucas Factorizations.

Column k=11 of A303217.

n=1;while(n<315,if(omega(fibonacci(n))==11,print1(n,", "));n++)

More terms from Ryan Propper, Apr 26 2006

a(52)-a(73) from Max Alekseyev, Aug 18 2013

A114837 Indices of Fibonacci numbers with 8 distinct prime factors.

Original entry on oeis.org

60, 72, 110, 112, 114, 128, 130, 135, 147, 154, 170, 171, 174, 217, 225, 231, 236, 238, 275, 279, 282, 290, 309, 316, 338, 355, 366, 374, 425, 436, 442, 452, 471, 481, 524, 538, 548, 553, 575, 642, 649, 694, 796, 801, 818, 833, 838, 847, 849, 851, 886, 889, 922, 923, 926, 939, 949, 958, 963, 965, 971, 979, 1037, 1041, 1077, 1079, 1094, 1111, 1127, 1137, 1141, 1153, 1154, 1189, 1211

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=60 because the 60th Fibonacci number consists of 8 distinct prime factors (i.e., 1548008755920 = 2^4 x 3^2 x 5 x 11 x 31 x 41 x 61 x 2521).

Amiram Eldar, Table of n, a(n) for n = 1..87
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=8 of A303217.

n=1;while(n<370,if(omega(fibonacci(n))==8,print1(n,", "));n++)

More terms from Ryan Propper, Apr 26 2006

a(53)-a(75) from Max Alekseyev, Aug 18 2013

cp's OEIS Frontend

A303215 A(n,k) is the n-th index of a Fibonacci number with exactly k prime factors (counted with multiplicity); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A114823 Indices of Fibonacci numbers with 13 distinct prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Extensions

A114836 Indices of Fibonacci numbers with 9 distinct prime factors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A114841 Indices of Fibonacci numbers with 3 distinct prime factors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Extensions

A114826 Indices of Fibonacci numbers with 10 distinct prime factors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A060320 Index of smallest Fibonacci number with exactly n distinct prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A303218 A(n,k) is the n-th Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs