A114826 -id:A114826 - cp's OEIS frontend

A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

3, 8, 4, 15, 9, 5, 20, 16, 10, 6, 30, 24, 18, 12, 7, 40, 36, 27, 21, 14, 11, 70, 48, 42, 28, 33, 19, 13, 60, 81, 54, 44, 32, 35, 22, 17, 80, 72, 104, 56, 45, 52, 37, 25, 23, 90, 84, 110, 105, 64, 50, 55, 38, 26, 29, 140, 126, 88, 112, 136, 78, 57, 74, 39, 31, 43

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   3,  8, 15, 20, 30,  40,  70,  60,  80,  90, ...
   4,  9, 16, 24, 36,  48,  81,  72,  84, 126, ...
   5, 10, 18, 27, 42,  54, 104, 110,  88, 165, ...
   6, 12, 21, 28, 44,  56, 105, 112,  96, 256, ...
   7, 14, 33, 32, 45,  64, 136, 114, 100, 258, ...
  11, 19, 35, 52, 50,  78, 148, 128, 108, 266, ...
  13, 22, 37, 55, 57,  92, 152, 130, 132, 296, ...
  17, 25, 38, 74, 63,  95, 164, 135, 138, 304, ...
  23, 26, 39, 77, 66,  99, 182, 147, 156, 322, ...
  29, 31, 46, 85, 68, 102, 186, 154, 184, 369, ...

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

Columns k=2-16 give: A114842, A114841, A114843, A114840, A114839, A114838, A114837, A114836, A114826, A114825, A114824, A114823, A117529, A117551, A117550.
Row n=1 gives: A060320.
Cf. A000045, A001221, A022307, A303215, A303218.

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; maxIndex = 200;
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k&];
T = Array[col, nmax];
A[n_, k_] := T[[k, n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2020 *)

A000045(A(n,k)) = A303218(n,k).

A001221(A000045(A(n,k))) = k.

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

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

A114838 Indices of Fibonacci numbers with 7 distinct prime factors.

Original entry on oeis.org

70, 81, 104, 105, 136, 148, 152, 164, 182, 186, 195, 207, 212, 244, 246, 254, 259, 289, 291, 292, 298, 305, 319, 326, 332, 344, 365, 367, 403, 404, 423, 445, 447, 451, 458, 478, 489, 511, 517, 519, 526, 533, 537, 543, 554, 565, 566, 597, 605, 679, 681, 685, 698, 699, 701, 721, 723, 725, 737, 745, 746, 749, 753, 758, 766, 767, 785, 813, 817, 831, 842, 871, 879, 901, 905, 914, 955, 967, 973, 985, 998, 1006, 1007, 1009, 1043, 1046, 1051, 1133, 1139, 1159, 1167, 1174, 1175, 1177, 1191, 1199, 1207, 1219

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=70 because the 70th Fibonacci number consists of 7 distinct prime factors (i.e., 190392490709135 = 5 x 11 x 13 x 29 x 71 x 911 x 141961).

Amiram Eldar, Table of n, a(n) for n = 1..112
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=7 of A303217.

Select[Range[1220],PrimeNu[Fibonacci[#]]==7&] (* Harvey P. Dale, Sep 18 2020 *)

n=1;while(n<310,if(omega(fibonacci(n))==7,print1(n,", "));n++)

More terms from Ryan Propper, Apr 26 2006

a(53)-a(98) from Max Alekseyev, Aug 18 2013

A114839 Indices of Fibonacci numbers with 6 distinct prime factors.

Original entry on oeis.org

40, 48, 54, 56, 64, 78, 92, 95, 99, 102, 116, 117, 129, 133, 155, 159, 175, 177, 188, 194, 205, 206, 219, 237, 245, 265, 278, 314, 323, 327, 339, 341, 343, 346, 356, 358, 361, 362, 394, 407, 411, 417, 422, 427, 437, 446, 454, 466, 482, 502, 503, 505, 514, 515, 527, 535, 542, 545, 551, 562, 573, 577, 583, 593, 607, 614, 622, 623, 625, 634, 655, 662, 667, 674, 713, 727, 731, 769, 781, 789, 791, 803, 809, 821, 835, 893, 917, 919, 974, 977, 982, 993, 995, 1013, 1039, 1047, 1057, 1081, 1097, 1103, 1121, 1138, 1151, 1165, 1172, 1202, 1203

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

Numbers n such that A000045(n) is in A046306.

			a(1) = 40 because 40th Fibonacci number consists of 6 distinct prime factors (i.e., 102334155 = 3 x 5 x 7 x 11 x 41 x 2161).
a(31) = 341 because F(341)= 89 * 557 * 2417 * 761227665342913 * 197907695243868721 * 4558282384863830955384586674337 has exactly 6 prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..123
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=6 of A303217.

n=1;while(n<330,if(omega(fibonacci(n))==6,print1(n,", "));n++)

More terms from Jonathan Vos Post, Mar 22 2006
Corrected by Ryan Propper, Apr 26 2006
a(55)-a(107) from Max Alekseyev, Aug 18 2013

A114840 Indices of Fibonacci numbers with 5 distinct prime factors.

Original entry on oeis.org

30, 36, 42, 44, 45, 50, 57, 63, 66, 68, 69, 75, 76, 98, 111, 118, 124, 134, 141, 153, 169, 172, 183, 185, 201, 202, 203, 213, 218, 229, 247, 253, 267, 302, 303, 329, 335, 347, 363, 371, 373, 377, 381, 382, 386, 395, 398, 413, 415, 439, 443, 461

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=30 because 30th Fibonacci number consists of 5 distinct prime factors (i.e., 832040 = 2^3 * 5 * 11 * 31 * 61).

Amiram Eldar, Table of n, a(n) for n = 1..116 (terms 1..106 from Max Alekseyev)
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=5 of A303217.

n=1;while(n<305,if(omega(fibonacci(n))==5,print1(n,", "));n++)

{n: A022307(n)=5}. - R. J. Mathar, Nov 29 2015

More terms from Ryan Propper, Apr 26 2006

a(56)-a(106) from Max Alekseyev, Aug 18 2013

A114842 Indices of Fibonacci numbers with 2 distinct prime factors.

Original entry on oeis.org

8, 9, 10, 12, 14, 19, 22, 25, 26, 31, 34, 41, 53, 59, 61, 71, 73, 79, 89, 94, 101, 107, 109, 113, 121, 127, 151, 167, 173, 191, 193, 199, 227, 251, 271, 277, 293, 331, 353, 397, 401, 467, 587, 599, 601, 613, 631, 653, 743, 991, 1091, 1223, 1373

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

A072381 is subsequence, since the only square Fibonacci numbers are 1 and 144 which are not squares of primes. - Charles R Greathouse IV, Sep 24 2012

			a(1) = 8 because 8th Fibonacci number consists of 2 distinct prime factors (i.e. 21 = 3*7).
25 is in the sequence because Fibonacci(25) = 75025 = 5^2 * 3001 consists of 2 distinct prime factors.

Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 2 (2005), 17-20.

Cf. A114823-A114826, A114836-A114841.

Column k=2 of A303217.

n=1;while(n<355,if(omega(fibonacci(n))==2,print1(n,", "));n++)

a(40)-a(50) from Donovan Johnson, Sep 27 2008
a(51)-a(52) from Max Alekseyev, Aug 18 2013
a(53) from Amiram Eldar, Oct 14 2019

A114843 Indices of Fibonacci numbers with 4 distinct prime factors.

Original entry on oeis.org

20, 24, 27, 28, 32, 52, 55, 74, 77, 85, 87, 91, 93, 97, 115, 123, 143, 146, 149, 157, 161, 163, 178, 187, 197, 209, 211, 214, 215, 221, 223, 239, 242, 249, 262, 269, 283, 287, 307, 311, 313, 321, 334, 349, 379, 391, 393, 409, 421, 453, 487, 493, 499, 523, 581, 586, 617, 641, 647, 677, 691, 707, 709, 787, 794, 811, 823, 839, 853, 859, 887, 907, 913, 929, 941, 953, 1031, 1049, 1063, 1093, 1229

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=20 because the 20th Fibonacci number consists of 4 distinct prime factors (i.e., 6765 = 3 x 5 x 11 x 41).

Amiram Eldar, Table of n, a(n) for n = 1..88
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=4 of A303217.

n=1;while(n<350,if(omega(fibonacci(n))==4,print1(n,", "));n++)

More terms from Ryan Propper, Apr 26 2006

a(56)-a(81) from Max Alekseyev, Aug 18 2013

A117551 Indices of Fibonacci numbers with 15 distinct prime factors.

Original entry on oeis.org

180, 210, 276, 280, 288, 312, 320, 340, 342, 416, 464, 476, 486, 516, 558, 564, 651, 665, 676, 708, 730, 735, 752, 776, 783, 837, 848, 856, 890, 992, 999, 1030, 1034, 1038, 1065, 1068, 1071, 1095, 1125, 1130, 1192, 1212

Offset: 1

Ryan Propper, Apr 26 2006

			F(180) = 2^4 * 3^3 * 5 * 11 * 17 * 19 * 31 * 41 * 61 * 107 * 181 * 541 * 2521 * 109441 * 10783342081 has 15 distinct prime factors, so 180 is in the sequence.

Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=15 of A303217.

isok(n) = (omega(fibonacci(n)) == 15) \\ Michel Marcus, Jun 28 2013

a(32)-a(42) from Max Alekseyev, Aug 18 2013

A117529 Indices of Fibonacci numbers with 14 distinct prime factors.

Original entry on oeis.org

168, 216, 294, 308, 348, 465, 530, 544, 657, 675, 824, 915, 927, 976, 1015, 1016, 1023, 1029, 1096, 1112, 1145, 1185

Offset: 1

Ryan Propper, Apr 26 2006

			F(168) = 2^5 * 3^2 * 7^2 * 13 * 23 * 29 * 83 * 167 * 211 * 281 * 421 * 1427 * 14503 * 65740583 has 14 distinct prime factors, so 168 is in the sequence.

Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A114823-A114826, A114836-A114841.

Column k=14 of A303217.

Position[PrimeNu[Fibonacci[Range[1200]]],14]//Flatten (* The program may take a long time to run *) (* Harvey P. Dale, Apr 27 2018 *)

a(15)-a(22) from Max Alekseyev, Aug 18 2013

cp's OEIS Frontend

A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A114841 Indices of Fibonacci numbers with 3 distinct prime factors.

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A114837 Indices of Fibonacci numbers with 8 distinct prime factors.

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A114838 Indices of Fibonacci numbers with 7 distinct prime factors.

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A114839 Indices of Fibonacci numbers with 6 distinct prime factors.

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A114840 Indices of Fibonacci numbers with 5 distinct prime factors.

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A114842 Indices of Fibonacci numbers with 2 distinct prime factors.

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A114843 Indices of Fibonacci numbers with 4 distinct prime factors.