A053409 -id:A053409 - cp's OEIS frontend

A064911 If n is semiprime (or 2-almost prime) then 1 else 0.

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Patrick De Geest, Oct 13 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65545 (first 10000 terms from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A010051, A064899-A064910, A053409, A046413, A101040, A105700, A280710, A302047, A302048, A302049, A353475, A353476, A353477, A353478, A353479, A353480, A353481.

a064911 = a010051 . a032742 -- Reinhard Zumkeller, Mar 13 2011

with(numtheory):
a:= n-> `if`(bigomega(n)=2, 1, 0):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 16 2011

Table[If[PrimeOmega[n] == 2, 1, 0], {n, 105}] (* Jayanta Basu, May 25 2013 *)

a(n)=bigomega(n)==2 \\ Charles R Greathouse IV, Mar 13 2011

a(n) = 1 iff n is in A001358 (semiprimes), a(n) = 0 iff n is in A100959 (non-semiprimes). - Reinhard Zumkeller, Nov 24 2004
Dirichlet g.f.: (primezeta(2s) + primezeta(s)^2)/2. - Franklin T. Adams-Watters, Jun 09 2006
a(n) = A057427(A174956(n)); a(n)*A072000(n) = A174956(n). - Reinhard Zumkeller, Apr 03 2010
a(n) = A010051(A032742(n)) (i.e., largest proper divisor is prime). - Reinhard Zumkeller, Mar 13 2011
From Antti Karttunen, Apr 24 2018 & Apr 22 2022: (Start)
a(n) = A280710(n) + A302048(n) = A101040(n) - A010051(n).
a(n) = A353478(n) + A353480(n) = A353477(n) + A353478(n) + A353479(n).
a(n) = A353475(n) + A353476(n).
(End)
a(n) = [Omega(n) = 2], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jul 22 2025

Edited by M. F. Hasler, Oct 18 2017

A072381 Numbers m such that Fibonacci(m) is a semiprime.

Original entry on oeis.org

8, 9, 10, 14, 19, 22, 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, 1487

Offset: 1

Shyam Sunder Gupta, Jul 20 2002

Note that there are two cases: (1) n is 2p, in which case the semiprime is Fibonacci(p)*Lucas(p) for some prime p, or (2) n is a power of a prime p^k for k > 0. In the first case, the primes p are in sequence A080327. In the second case, it appears that k=1 except for n = 8, 9 and 121. - T. D. Noe, Sep 23 2005
The associated sequence of Fibonacci numbers contains no squares, since the only Fibonacci numbers which are square are 1 and 144. Consequently this is a subsequence of A114842. - Charles R Greathouse IV, Sep 24 2012
Sequence continues as 1543?, 1709, 1741?, 1759, 1801?, 1889, 1987, ..., where ? marks uncertain terms. - Max Alekseyev, Jul 10 2016

			a(4) = 14 because the 14th Fibonacci number 377 = 13*29 is a semiprime.

Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
Ron Knott, Fibonacci numbers
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A053409, A085726 (n such that n-th Lucas number is a semiprime).

Column k=2 of A303215.

Select[Range[200], Plus@@Last/@FactorInteger[Fibonacci[ # ]] == 2&] (Noe)
Select[Range[1500],PrimeOmega[Fibonacci[#]]==2&] (* Harvey P. Dale, Dec 13 2020 *)

for(n=2,9999,bigomega(fibonacci(n))==2&&print1(n",")) \\ - M. F. Hasler, Oct 31 2012

issemi(n)=bigomega(n)==2
is(n)=if(n%2, my(p); if(issquare(n,&p), isprime(p) && isprime(fibonacci(p)) && isprime(fibonacci(n)/fibonacci(p)), isprime(n) && issemi(fibonacci(n))), (isprime(n/2) && isprime(fibonacci(n/2)) && isprime(fibonacci(n)/fibonacci(n/2))) || n==8) \\ Charles R Greathouse IV, Oct 06 2016

More terms from Don Reble, Jul 31 2002
a(49)-a(50) from Max Alekseyev, Aug 18 2013
a(51)-a(52) from Max Alekseyev, Jul 10 2016

A303216 A(n,k) is the n-th Fibonacci number with exactly k prime factors (counted with multiplicity); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

2, 21, 3, 8, 34, 5, 6765, 610, 55, 13, 2584, 196418, 987, 377, 89, 144, 701408733, 317811, 10946, 4181, 233, 832040, 102334155, 1134903170, 2178309, 75025, 17711, 1597, 86267571272, 267914296, 12586269025, 365435296162, 32951280099, 3524578, 121393, 28657

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
    2,    21,       8,         6765,           2584,                 144, ...
    3,    34,     610,       196418,      701408733,           102334155, ...
    5,    55,     987,       317811,     1134903170,         12586269025, ...
   13,   377,   10946,      2178309,   365435296162,      10610209857723, ...
   89,  4181,   75025,  32951280099,  6557470319842,    2111485077978050, ...
  233, 17711, 3524578, 139583862445, 72723460248141, 7540113804746346429, ...

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

Columns k=1-2 give: A005478, A053409.
Row n=1 gives A072397.
Cf. A000045, A001222, 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))

A[n_, k_] := Module[{F = Fibonacci, h, p, q = 2}, p[_] = {}; While[ Length[p[k]] < n, q = q+1; h = PrimeOmega[F[q]]; p[h] = Append[p[h], F[q]]]; p[k][[n]]];
Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Feb 05 2021, after Alois P. Heinz *)

A(n,k) = A000045(A303215(n,k)).

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

A064910 Smallest semiprime p*q such that q >= p and q mod p = n.

Original entry on oeis.org

4, 6, 15, 65, 77, 133, 91, 319, 209, 341, 299, 481, 493, 799, 527, 1007, 1139, 2449, 703, 3611, 989, 1541, 1643, 3589, 1537, 2407, 2747, 2759, 1829, 3811, 1891, 4633, 2993, 3959, 2627, 4033, 2701, 6157, 3239, 9073, 3569, 5461, 4183, 6439, 5141, 6533

Offset: 0

Patrick De Geest, Oct 13 2001

John Cerkan, Table of n, a(n) for n = 0..10000
John Cerkan, Python 2.7 program

Cf. A001358 (p2 mod p1 = 0), A064899-A064909, A064911, A053409, A046413.

nsp[n_Integer] := nsp[n] = Block[{sp = n + 1}, While[PrimeOmega[sp] != 2, sp++]; sp]; a[n_Integer] := Block[{sp = 4}, While[ fi = FactorInteger@ sp; Mod[fi[[-1, 1]], fi[[1, 1]]] != n, sp = nsp[sp]]; sp]; Array[a, 46, 0] (* Robert G. Wilson v, Aug 20 2025 *)

Name amended by John Cerkan, Apr 12 2018

A122498 Padovan numbers that are semiprimes.

Original entry on oeis.org

4, 9, 21, 49, 65, 86, 265, 1081, 1897, 2513, 7739, 97229, 128801, 299426, 922111, 1221537, 2839729, 62608681, 338356945, 53406819691, 2066337330754, 6363483400447, 8429820731201, 432062194544201, 7190854504969591, 12619069972000553, 16716708595637087

Offset: 1

Roger L. Bagula, Sep 15 2006

nonn

The smallest candidate for the next term in the b-file is A000931(1958), which is composite with 239 digits and an unknown number of prime factors. - Hugo Pfoertner, Sep 07 2017

Hugo Pfoertner, Table of n, a(n) for n = 1..54
Eric Weisstein's World of Mathematics, Semiprime

Cf. A000931, A001358, A053409, A100891.

select(x-> numtheory[bigomega](x)=2, [(<<0|1|0>,
   <0|0|1>, <1|1|0>>^i)[1$2]$i=0..300])[];  # Alois P. Heinz, Aug 31 2017

SemiprimeQ[1] := False SemiprimeQ[n_Integer] := Plus @@ (Last /@ FactorInteger[n]) == 2 a = Table[ SeriesCoefficient[ Series[x/(1 - x^2 - x^3), {x, 0, 50}], n], {n, 0, 50}] f[n_] = If[SemiprimeQ[a[[n]]] == True, a[[n]], {}] Flatten[Table[f[n], {n, 1, Length[a]}]]

More terms from Alois P. Heinz, Aug 31 2017

A137563 Fibonacci numbers with three distinct prime divisors.

Original entry on oeis.org

610, 987, 2584, 10946, 3524578, 9227465, 24157817, 39088169, 63245986, 1836311903, 7778742049, 20365011074, 591286729879, 4052739537881, 17167680177565, 44945570212853, 61305790721611591, 420196140727489673, 1500520536206896083277, 6356306993006846248183

Offset: 1

Parthasarathy Nambi, Apr 25 2008

nonn

			The distinct prime divisors of the Fibonacci number 610 are 2, 5 and 61.
The distinct prime divisors of the Fibonacci number 44945570212853 are 269, 116849 and 1429913.

Michel Marcus and Amiram Eldar, Table of n, a(n) for n = 1..83 (terms 1..80 from Michel Marcus)
Ron Knott, Fibonacci Numbers.
Ron Knott, Prime divisors of Fibonacci numbers.

Cf. A053409, A114841.
Intersection of A033992 and A000045. - Michel Marcus, Mar 24 2018
Column k=3 of A303218.

P1:=List([1..110],n->Fibonacci(n));;
P2:=List([1..Length(P1)],i->Filtered(DivisorsInt(P1[i]),IsPrime));;
a:=List(Filtered([1..Length(P2)],i->Length(P2[i])=3),j->P1[j]); # Muniru A Asiru, Mar 25 2018

with(numtheory): with(combinat): a:=proc(n) if nops(factorset(fibonacci(n)))= 3 then fibonacci(n) else end if end proc: seq(a(n),n=1..110); # Emeric Deutsch, May 18 2008

Select[Array[Fibonacci, 120], PrimeNu@ # == 3 &] (* Michael De Vlieger, Apr 10 2018 *)

lista(nn) = for (n=1, nn, if (omega(f=fibonacci(n))==3, print1(f, ", "))); \\ Michel Marcus, Mar 24 2018

a(n) = A000045(A114841(n)). - Michel Marcus, Mar 24 2018

More terms from Emeric Deutsch, May 18 2008

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);

A099954 Numbers k such that Fibonacci(k) and its reversal are two distinct semiprimes.

Original entry on oeis.org

19, 22, 31, 41, 59, 107, 193, 199, 227, 467

Offset: 1

Jonathan Vos Post and G. L. Honaker, Jr., Nov 13 2004

a(11) > 1000. - Donovan Johnson, Jun 06 2009

a(11) >= 1801. Inclusion of 1801 depends on the factorization of Fibonacci(1801), a 377-digit composite number. - Tyler Busby, Jan 14 2023

			F(19) = 4181 = 37 * 113, reverse(F(19)) = 1814 = 2 * 907.

factordb, Status of Fibonacci(1801).

F(a(n)) is the intersection of A053409 and A097393

with(combinat): with(numtheory): rev:=proc(n) local nn: nn:=convert(n,base,10): add(nn[nops(nn)+1-j]*10^(j-1),j=1..nops(nn)) end: a:=proc(n): if rev(fibonacci(n))<>fibonacci(n) and bigomega(fibonacci(n))=2 and bigomega(rev(fibonacci(n)))=2 then n else fi end: seq(a(n),n=1..200); # Emeric Deutsch, Jul 26 2006

fspQ[n_]:=Module[{f=Fibonacci[n]},f!=IntegerReverse[f]&&PrimeOmega[f] == PrimeOmega[IntegerReverse[f]]==2]; Select[Range[470],fspQ] (* Harvey P. Dale, Jul 24 2016 *)

is(k) = {(fib=fibonacci(k))!=(fibrev=fromdigits(Vecrev(digits(fib)))) && (bigomega(fib)==2 && bigomega(fibrev)==2)} \\ Tyler Busby, Jan 07 2023

More terms from Emeric Deutsch, Jul 26 2006

a(10) from Donovan Johnson, Jun 06 2009

A136341 Fibonacci primes or semiprimes F(k) such that F(k+1) is again prime or semiprime.

Original entry on oeis.org

2, 3, 13, 21, 34, 55, 233, 17711

Offset: 1

Cino Hilliard, Mar 28 2008

By definition, the smaller number in a pair of two consecutive Fibonacci numbers in A061305. a(9), if it exists, is >= A000045(230), so it has at least 48 digits. [R. J. Mathar, Feb 06 2010]

A search for consecutive numbers in the union of A072381 and A001605 shows that a(9) must be larger than A000045(990), a number with 207 digits, if it exists. [R. J. Mathar, Jun 02 2010]

			(55,89) is an almost twin Fibonacci prime pair because 55=5*11 is a 2-almost prime and 89 is prime.

Cf. A001358.
Cf. A053409, A005478. [R. J. Mathar, Jun 02 2010]
Cf. A001605, A072381, A278637.

Fibonacci[#]&/@(SequencePosition[Table[If[PrimeOmega[f]<=2,1,0],{f,Fibonacci[ Range[150]]}],{1,1}][[All,1]]) (* Harvey P. Dale, Mar 29 2022 *)

ATfib(n) = for(x=3,n,f1=fibonacci(x);f2=fibonacci(x+1);if(bigomega (f1)<=2&&bigomega(f2)<=2, print1(f1",")))

for( k=3,10^5, bigomega( fibonacci( k++ ))>2 & next; bigomega( fibonacci( k-1 ))>2 & next; print1(fibonacci(k--)",")) \\ M. F. Hasler, May 01 2008

Let F(n) = n-th Fibonacci number and define a 2-almost prime number to be a number with only 2 prime divisors with multiplicity.

Edited by M. F. Hasler, May 01 2008

A173684 Semiprimes of the form Fibonacci(k) + k.

Original entry on oeis.org

10, 14, 65, 391, 1003, 2602, 10967, 2178341, 701408777, 86267571326, 591286729937, 4052739537943, 72723460248209, 117669030461063, 3416454622906783, 61305790721611673, 420196140727489759, 2427893228399975082557, 251728825683549488150424389

Offset: 1

Jonathan Vos Post, Jan 26 2011

nonn

This is to A069108 as semiprimes are to primes. A002062(k) is semiprime for k = 5, 6, 10, 14, 16, 18, 21, 32, ...

			F(21) + 21 = 10967 = 11 * 997, thus 10967 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..24

Cf. A000045, A001358, A002062, A053409, A069108.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [ a: n in [1..100] | IsSemiprime(a) where a is n+Fibonacci(n) ]; // Klaus Brockhaus, Jan 27 2011

Select[Table[Fibonacci[n]+n,{n,200}],PrimeOmega[#]==2&] (* Harvey P. Dale, Nov 02 2011 *)

A002062 INTERSECTION A001358.

cp's OEIS Frontend

A064911 If n is semiprime (or 2-almost prime) then 1 else 0.

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A072381 Numbers m such that Fibonacci(m) is a semiprime.

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A303216 A(n,k) is the n-th 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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A064910 Smallest semiprime p*q such that q >= p and q mod p = n.

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A122498 Padovan numbers that are semiprimes.

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Mathematica

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A137563 Fibonacci numbers with three distinct prime divisors.

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GAP

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

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