A072381 -id:A072381 - 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.

A053409 Fibonacci numbers which are semiprimes.

Original entry on oeis.org

21, 34, 55, 377, 4181, 17711, 121393, 1346269, 5702887, 165580141, 53316291173, 956722026041, 2504730781961, 308061521170129, 806515533049393, 14472334024676221, 1779979416004714189, 19740274219868223167, 573147844013817084101, 10284720757613717413913

Offset: 1

G. L. Honaker, Jr., Jan 09 2000

nonn

Subsequence of A006881, since the only square Fibonacci numbers are 1 and 144. - Charles R Greathouse IV, Sep 24 2012

Apart from a(1) = 21, all terms are of the form F(p), F(2p), or F(p^2) where F(n) is the n-th Fibonacci number. - Charles R Greathouse IV, Oct 06 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..48

Cf. A000045, A001358, A006881, A072381.

Column k=2 of A303216.

Select[Fibonacci@Range[120],Last/@FactorInteger[#]=={1,1}&] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
Select[Fibonacci[Range[150]],PrimeOmega[#]==2&] (* Harvey P. Dale, Jun 26 2020 *)

issemi(n)=bigomega(n)==2
list(lim)=my(v=List([21]),F,t); forprime(p=2,, F=fibonacci(p); if(F>lim, break); if(issemi(F), listput(v,F))); forprime(p=2,, F=fibonacci(p^2); if(F>lim, break); if(isprime(t=fibonacci(p)) && isprime(F/t), listput(v,F))); forprime(p=2,, F=fibonacci(2*p); if(F>lim, break); if(isprime(t=fibonacci(p)) && isprime(F/t), listput(v,F))); Set(v) \\ Charles R Greathouse IV, Oct 06 2016

a(n) = A000045(A072381(n)).

A278245 Least number with the same prime signature as the n-th Fibonacci number: a(n) = A046523(A000045(n)).

Original entry on oeis.org

1, 1, 2, 2, 2, 8, 2, 6, 6, 6, 2, 144, 2, 6, 30, 30, 2, 120, 6, 210, 30, 6, 2, 10080, 12, 6, 210, 210, 2, 9240, 6, 210, 30, 6, 30, 166320, 30, 30, 30, 30030, 6, 9240, 2, 2310, 2310, 30, 2, 2882880, 30, 4620, 30, 210, 6, 120120, 210, 60060, 2310, 30, 6, 232792560, 6, 30, 2310, 30030, 30, 9240, 30, 2310, 2310, 510510, 6, 1396755360, 6, 210, 4620, 2310, 210, 120120, 6

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used as a filter for certain sequences involving Fibonacci numbers as it matches to any sequence that is obtained as f(A000045(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

			From _Michael De Vlieger_, May 18 2017: (Start)
a(6) = 8 because Fibonacci(6) = 8, the multiplicity of the prime factor of 8 is 3; the smallest p^3 = 2^3 = 8.
a(7) = 2 because Fibonacci(7) = 13, the multiplicity of the prime factor of 13 is 1; the smallest p^1 = 2^1 = 2.
a(15) = 30 because Fibonacci(15) = 610. The multiplicities of the prime factors of 610, in order from greatest to least, are {1, 1, 1}, the smallest prime power product p^1 * q^1 * r^1 = 2 * 3 * 5 = 30.
a(18) = 120 because Fibonacci(18) = 2584 = 2^3 * 17 * 19 -> 2^3 * 3 * 5 = 120. (End)

Antti Karttunen (terms 1..374) & Hans Havermann, Table of n, a(n) for n = 1..1300
Index entries for sequences computed from exponents in factorization of n

Cf. A000045, A046523, A278241, A278248.
Cf. A286545 (rgs-version of this sequence), A286467.
Cf. A001605 (positions of 2's), A072381 (of 6's).
Sequences with matching equivalence classes: A063375, A105307, A152774.

Table[If[# == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &,
Sort[FactorInteger[#][[All, -1]], Greater]]] &@ Fibonacci@ n, {n, 79}] (* Michael De Vlieger, May 18 2017 *)

A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
f0 = 0; f1 = 1; for(n=1, 10000, write("b278245.txt", n, " ", A046523(f1)); old_f0 = f0; f0 = f1; f1 = f1 + old_f0; );

(define (A278245 n) (A046523 (A000045 n)))

a(n) = A046523(A000045(n)).

A241483 Primes p such that p+2, p+4, p+6, p+8, p+10 and p+12 are all semiprime.

Original entry on oeis.org

1381, 3089, 10399, 49081, 53759, 63949, 76801, 98491, 107509, 109397, 113341, 143093, 182747, 204331, 209477, 239087, 252949, 255989, 313409, 396983, 426287, 500341, 602779, 677333, 812281, 832801, 1516531, 1574939, 1599151, 1619507, 1678639, 1866737, 2046449

Offset: 1

K. D. Bajpai, Apr 23 2014

nonn

			1381 is prime and appears in the sequence because 1381+2 = 1383 = 3*461, 1381+4 = 1385 = 5*277, 1381+6 = 1387 = 19*73, 1381+8 = 1389 = 3*463, 1381+10 = 1391 = 13*107 and  1381+12 = 1393 = 7*199, which are all semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..510

Cf. A072381, A082919.

with(numtheory): KD:= proc() local a,b,d,e,f,g,k; k:=ithprime(n); a:=bigomega(k+2); b:=bigomega(k+4); d:=bigomega(k+6);  e:=bigomega(k+8); f:=bigomega(k+10); g:=bigomega(k+12);  if a=2 and  b=2 and  d=2 and  e=2 and  f=2 and  g=2then RETURN (k);  fi; end: seq(KD(), n=1..200000);

KD = {};  Do[t = Prime[n]; If[PrimeOmega[t + 2] == 2 && PrimeOmega[t + 4] == 2 && PrimeOmega[t + 6] == 2 && PrimeOmega[t + 8] == 2 && PrimeOmega[t + 10] == 2 && PrimeOmega[t + 12] == 2, AppendTo[KD, t]], {n, 200000}]; KD
Select[Prime[Range[155000]],Union[PrimeOmega/@(#+2Range[6])]=={2}&] (* Harvey P. Dale, Dec 13 2018 *)

is(n)=if(n%3==1, isprime((n+2)/3) && isprime((n+8)/3) && bigomega(n+4)==2 && bigomega(n+10)==2, isprime((n+4)\3) && isprime((n+10)\3) && bigomega(n+2)==2 && bigomega(n+8)==2) && isprime(n) && bigomega(n+6)==2 && bigomega(n+12)==2
forprime(p=2,1e7,if(is(p),print1(p", "))) \\ Charles R Greathouse IV, Aug 25 2014

A227875 Fibonacci numbers which are perfect powers.

Original entry on oeis.org

0, 1, 8, 144

Offset: 1

Jean-François Alcover, Oct 25 2013

Also, Fibonacci numbers which are products of Fibonacci numbers (each greater than 1 when the product is greater than 1 - see A235383). - Rick L. Shepherd, Feb 19 2014
The terms of the subsequence (1, 8, 144) are the Fibonacci numbers that are powerful numbers. - Robert C. Lyons, Jul 12 2016
Also Fibonacci numbers without any primitive divisors. See [Heuberger & Wagner]. - Michel Marcus, Aug 21 2016
It was proved (Bugeaud, Mignotte, and Siksek, 2006, p. 971) that the only perfect powers among the Fibonacci numbers and Lucas numbers are {0, 1, 8, 144} and {1, 4}, respectively. - Daniel Forgues, Apr 09 2018

Vladica Andrejic, On Fibonacci Powers, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 17 (2006), 38-44.
Yann Bugeaud, Florian Luca, Maurice Mignotte, and Samir Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
Yann Bugeaud, Maurice Mignotte, and Samir Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Annals of Mathematics, 163 (2006), pp. 969-1018.
Clemens Heuberger and Stephan Wagner, On the monoid generated by a Lucas sequence, arXiv:1606.02639 [math.NT], 2016. Gives the complement sequence w.r.t Fibonacci numbers.
J. Mc Laughlin, Small prime powers in the Fibonacci sequence, arXiv:math/0110150 [math.NT] (2001).
Attila Pethő, Diophantine properties of linear recursive sequences II, Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 17:2 (2001), pp. 81-96.

Cf. A000045, A001597, A072381, A114842.

perfectPowerQ[0] = True; perfectPowerQ[1] = True; perfectPowerQ[n_] := GCD @@ FactorInteger[n][[All, 2]] > 1; Union[Select[Fibonacci /@ Range[0, 20], perfectPowerQ]]

A085726 Numbers n such that n-th Lucas number is a semiprime.

Original entry on oeis.org

3, 10, 14, 20, 23, 26, 29, 32, 38, 43, 49, 56, 62, 64, 67, 68, 73, 76, 80, 83, 86, 89, 97, 107, 121, 128, 136, 137, 157, 164, 167, 172, 178, 197, 202, 211, 223, 229, 284, 293, 307, 311, 328, 373, 389, 397, 458, 487, 521, 541, 557, 577, 586, 619, 673, 857, 914, 929, 947, 1082, 1151, 1249, 1277, 1279, 1306, 1318, 1493, 1499, 1667

Offset: 1

Jason Earls, Jul 20 2003

nonn

From results on the divisibility of generalized Fibonacci sequences (2nd order recurrences with various integer initial values), it follows that if n is such that n-th Lucas number is a semiprime, it is necessary but not sufficient that n have at most two distinct prime factors (A070915). That is: A000204(n) an element of A001358 implies n an element of A070915. - Jonathan Vos Post, Sep 22 2005
All numbers in this sequence have the form 2^r p^s, where p is an odd prime and r and s are not both zero. It appears that s=2 for only p=7 and 11, otherwise s=0 or 1. - T. D. Noe, Nov 29 2005
Sequence continues as 1831?, 1877?, 1901, 1951, ... where ? mark uncertain terms. - Max Alekseyev, Aug 18 2013

Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000204.

Cf. A072381 (n such that Fibonacci(n) is a semiprime).

IsSemiprime:=func; [n: n in [2..300] | IsSemiprime(Lucas(n))]; // Vincenzo Librandi, Feb 12 2016

a = 1; b = 3; Do[c = a + b; If[Plus@@Last/@FactorInteger[c] == 2, Print[n]]; a = b; b = c, {n, 3, 200}] (* Ryan Propper, Jun 28 2005 *)
Select[Range[400], PrimeOmega[LucasL[#]] == 2 &] (* Vincenzo Librandi, Feb 12 2016 *)

isok(n) = bigomega(fibonacci(n+1)+fibonacci(n-1)) == 2; \\ Michel Marcus, Feb 12 2016

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Aug 25 2004
More terms from Ryan Propper, Jun 28 2005
More terms from T. D. Noe, Nov 29 2005
a(60)-a(62) from Max Alekseyev, Aug 18 2013
a(63)-a(69) from Sean A. Irvine, Feb 11 2016

A241493 Primes p such that p + 4, p + 16, p + 64, p + 256 and p + 1024 are all semiprimes.

Original entry on oeis.org

1627, 2917, 3583, 4603, 5581, 6367, 6379, 8263, 9697, 12517, 12763, 13339, 14197, 15289, 16339, 16993, 17539, 17737, 18199, 19267, 19531, 20023, 28057, 28879, 29587, 32647, 33427, 34033, 34537, 35353, 35617, 37039, 37087, 37657, 37663, 42337, 43093, 47533, 48049

Offset: 1

K. D. Bajpai, Apr 24 2014

nonn

The constants in the definition (4, 16, 64, 256 and 1024 ) are in geometric progression.

			1627 is prime and appears in the sequence because 1627+4 = 1631 = 7*233, 1627+16 = 1643 = 31*53, 1627+64 = 1691 = 19*89, 1627+256 = 1883 = 7*269 and 1627+1024 = 2651 = 11*241, which are all semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A072381, A082919, A241483, A241484.

with(numtheory): KD:= proc() local a,b,d,e,f,k; k:=ithprime(n); a:=bigomega(k+4); b:=bigomega(k+16); d:=bigomega(k+64); e:=bigomega(k+256); f:=bigomega(k+1024); if a=2 and  b=2 and d=2 and  e=2 and f=2 then RETURN (k); fi; end: seq(KD(), n=1..10000);

KD = {}; Do[t = Prime[n]; If[PrimeOmega[t + 4] == 2 && PrimeOmega[t + 16] == 2 && PrimeOmega[t + 64] == 2 && PrimeOmega[t + 256] == 2 && PrimeOmega[t + 1024] == 2, AppendTo[KD, t]], {n, 10000}]; KD
(* For the b-file *) c = 0; Do[t = Prime[n]; If[PrimeOmega[t + 4] == 2 && PrimeOmega[t + 16] == 2 && PrimeOmega[t + 64] == 2 && PrimeOmega[t + 256] == 2 && PrimeOmega[t + 1024] == 2, c++; Print[c, "  ", t]], {n, 1,5*10^6}];
Select[Prime[Range[5000]],Union[PrimeOmega[#+{4,16,64,256,1024}]] == {2}&] (* Harvey P. Dale, Nov 28 2017 *)

A286467 Compound filter (prime signature of n & prime signature of the n-th Fibonacci number): a(n) = P(A101296(n), A286545(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 5, 9, 5, 19, 5, 33, 18, 25, 5, 51, 5, 25, 40, 73, 5, 72, 12, 84, 40, 25, 5, 128, 69, 25, 71, 84, 5, 180, 12, 146, 40, 25, 40, 242, 23, 40, 40, 198, 12, 180, 5, 177, 177, 40, 5, 337, 31, 216, 40, 84, 12, 284, 59, 308, 140, 40, 12, 478, 12, 40, 177, 339, 40, 180, 23, 177, 140, 387, 12, 610, 12, 59, 216, 177, 59, 309, 12, 540, 332, 40, 5, 608, 59, 40, 59

Offset: 1

Antti Karttunen, May 17 2017

nonn

Nonsquare semiprimes pq for which F(pq) is also a semiprime is given by the positions where 25's occur in this sequence: 10, 14, 22, 26, 34, 94, (any more terms?). This is a subsequence of A072381.

Antti Karttunen, Table of n, a(n) for n = 1..1300 (based also on the b-file of A278245 provided by Hans Havermann)

Cf. A000027, A046523, A072381, A101296, A278245, A286545.

Cf. A083668 (positions of 5's).

(define (A286467 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286545 n)) 2) (- (A101296 n)) (- (* 3 (A286545 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n) + A286545(n))^2) - A101296(n) - 3*A286545(n)).

A075735 Squarefree Fibonacci numbers with an even number of prime factors (mu(n)=1).

Original entry on oeis.org

1, 1, 21, 34, 55, 377, 4181, 6765, 17711, 121393, 196418, 317811, 1346269, 2178309, 5702887, 102334155, 165580141, 32951280099, 53316291173, 139583862445, 956722026041, 2504730781961, 10610209857723, 308061521170129

Offset: 1

Jani Melik, Oct 07 2002

			21 is a Fibonacci number and 21=3*7, 34 is a Fibonacci numbers and 34=2*17, ...

Amiram Eldar, Table of n, a(n) for n = 1..505 (terms 1..100 from T. D. Noe)

Subsequence of A061305 (squarefree Fibonacci numbers).
Cf. A000045, A030229, A074691.
Cf. A022307, A053409, A072381.

with(combinat, fibonacci): m1_fib := proc(n); if (numtheory[mobius](fibonacci(n))=1) then RETURN(fibonacci(n)); fi; end: seq(m1_fib(i), i=1..100);

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

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.

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A053409 Fibonacci numbers which are semiprimes.

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A278245 Least number with the same prime signature as the n-th Fibonacci number: a(n) = A046523(A000045(n)).

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A241483 Primes p such that p+2, p+4, p+6, p+8, p+10 and p+12 are all semiprime.

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A227875 Fibonacci numbers which are perfect powers.

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A085726 Numbers n such that n-th Lucas number is a semiprime.

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A241493 Primes p such that p + 4, p + 16, p + 64, p + 256 and p + 1024 are all semiprimes.

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A286467 Compound filter (prime signature of n & prime signature of the n-th Fibonacci number): a(n) = P(A101296(n), A286545(n)), where P(n,k) is sequence A000027 used as a pairing function.