A072381 -id:A072381 - cp's OEIS frontend

A140987 Number of groups of order F(n).

1, 1, 1, 1, 1, 5, 1, 2, 2, 2, 1, 197, 1, 1, 6, 2, 1, 41, 1, 7, 4, 1, 1

Offset: 1

Jonathan Vos Post, Jul 28 2008

			a(6) = 5 because Fibonacci number F(6) = 8 and there are 5 groups of order 8.

GAP package GrpConst

Cf. A000001, A000040, A000045, A001358, A001605, A072381.

a(n) = A000001(A000045(n)).

Entries through a(17) checked by N. J. A. Sloane, Aug 01 2008

a(18)-a(23) from Eric M. Schmidt, Jun 19 2014

A241484 Primes p such that p+2 and p+4 are semiprime.

Original entry on oeis.org

2, 31, 47, 53, 83, 89, 139, 157, 181, 199, 211, 233, 263, 317, 337, 389, 409, 443, 449, 467, 541, 577, 587, 631, 677, 683, 719, 751, 787, 811, 839, 919, 947, 991, 1039, 1097, 1117, 1163, 1187, 1201, 1259, 1367, 1381, 1399, 1559, 1637, 1669, 1709, 1759, 1777, 1847

Offset: 1

K. D. Bajpai, Apr 23 2014

nonn

			31 is prime and appears in the sequence because 31+2 = 33 = 3*11 and 31+4 = 35 = 5*7, which are semiprime.
53 is prime and appears in the sequence because 53+2 = 55 = 5*11 and 53+4 = 57 = 3*19, which are semiprime.

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

Cf. A072381, A082919.

IsSemiprime:=func< p | &+[ k[2]: k in Factorization(p)] eq 2 >; [p: p in PrimesUpTo(2000)| IsSemiprime(p+2) and IsSemiprime(p+4)]; // Vincenzo Librandi, Apr 24 2014

with(numtheory): KD:= proc() local a,b,d,k; k:=ithprime(n); a:=bigomega(k+2);b:=bigomega(k+4); if a=2 and  b=2 then RETURN (k); fi; end: seq(KD(), n=1..1000);

KD = {}; Do[t = Prime[n];If[PrimeOmega[t + 2] == 2 && PrimeOmega[t + 4] == 2,AppendTo[KD, t]], {n, 1000}]; KD

A241554 Semiprimes generated by the polynomial 2 * n^2 + 29.

Original entry on oeis.org

1711, 1829, 2077, 2479, 3071, 3901, 5029, 6527, 6757, 7471, 7967, 8479, 10397, 10981, 11581, 14141, 15167, 15517, 15871, 16591, 16957, 17701, 18079, 18847, 19631, 20837, 22927, 23791, 25567, 26941, 27877, 28829, 29797, 30287, 31279, 31781, 32287, 35941, 38117

Offset: 1

K. D. Bajpai, Apr 25 2014

nonn

2 * n^2 + 29 is a well-known Legendre prime-producing polynomial which generates 29 distinct primes for n = 0, 1, ..., 28. For n = 29, it yields the first semiprime, 1711 = 29 * 59.

The number n = 185 is the least positive integer for which 2*n^2 + 29 = 68479 = 31 * 47 * 47 is not squarefree.

			2 * 30^2 + 29 = 1829 = 31 * 59, which is a semiprime and is a term.
2 * 35^2 + 29 = 2479 = 37 * 67, which is a semiprime and is a term.

K. D. Bajpai, Table of n, a(n) for n = 1..10000
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Cf. A007641 (for primes).

Cf. A001358, A072381, A082919, A145292, A228183, A237627, A353004, A353388.

with(numtheory):A241554:= proc() local k; k:=2*x^2+29;if bigomega(k)=2 then RETURN (k); fi; end: seq(A241554(), x=0..500);

A241554 = {}; Do[k = 2 * n^2 + 29; If[PrimeOmega[k] == 2, AppendTo[A241554, k]], {n,200}]; A241554

s=[]; for(n=1, 200, t=2*n^2+29; if(bigomega(t)==2, s=concat(s, t))); s \\ Colin Barker, Apr 26 2014

A286545 Restricted growth sequence of A278245 (prime signature of Fibonacci numbers).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 4, 4, 4, 2, 5, 2, 4, 6, 6, 2, 7, 4, 8, 6, 4, 2, 9, 10, 4, 8, 8, 2, 11, 4, 8, 6, 4, 6, 12, 6, 6, 6, 13, 4, 11, 2, 14, 14, 6, 2, 15, 6, 16, 6, 8, 4, 17, 8, 18, 14, 6, 4, 19, 4, 6, 14, 13, 6, 11, 6, 14, 14, 20, 4, 21, 4, 8, 16, 14, 8, 17, 4, 22, 20, 6, 2, 23, 8, 6, 8, 22, 4, 24, 25, 13, 8, 4, 13, 26, 8, 14, 13, 27, 4, 17, 6, 20, 20, 6, 4, 28

Offset: 1

Antti Karttunen, May 17 2017

nonn

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

Cf. A278245, A286467.

Cf. A001605 (positions of 2's), A072381 (of 4's).

A114812 Indices of Fibonacci numbers with 3 prime factors when counted with multiplicity.

Original entry on oeis.org

6, 15, 16, 21, 25, 33, 35, 37, 38, 39, 46, 49, 51, 58, 62, 65, 67, 82, 86, 103, 106, 119, 122, 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

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

1811, 1933, 1997, 2069, 2087, 2203, 2221, 2311, 2663, 2713, 3631, 4157, 4651, 5107, 6701, 7211, 8123 are also terms (from data in Kelly link). - Chai Wah Wu, Nov 11 2019

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

Amiram Eldar, Table of n, a(n) for n = 1..83
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A072381, A114813.

Column k=3 of A303215.

t = {}; Do[f = FactorInteger[Fibonacci[n]]; If[Total[Transpose[f][[2]]] == 3, AppendTo[t, n]], {n, 2, 100}]; t (* T. D. Noe, Mar 14 2014 *)
Flatten[Position[Fibonacci[Range[700]],?(PrimeOmega[#]==3&)]] (* _Harvey P. Dale, Feb 15 2015 *)

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

{n: A038575(n)=3}. [R. J. Mathar, Jun 08 2010]

More terms from Ryan Propper, May 22 2006

A241716 Primes p such that p^3 - 2 is semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 41, 43, 47, 61, 79, 89, 101, 107, 139, 157, 181, 199, 239, 271, 307, 311, 331, 337, 347, 349, 379, 397, 409, 421, 431, 479, 487, 499, 521, 523, 541, 571, 607, 613, 641, 643, 661, 673, 701, 719, 761, 769, 811, 823, 829, 839, 877, 881, 883

Offset: 1

K. D. Bajpai, Apr 27 2014

nonn

			11 is prime and appears in the sequence because 11^3 - 2 = 1329 = 3 * 443, which is a semiprime.
17 is prime and appears in the sequence because 17^3 - 2 = 4911 = 3 * 1637, which is a semiprime.
23 is prime but does not appear in the sequence because 23^3 - 2 = 12165 =  3 * 5 * 811, which is not a semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..9380

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

with(numtheory):A241716:= proc() local k; k:=ithprime(x); if bigomega(k^3-2)=2 then RETURN (k); fi; end: seq(A241716(), x=1..500);

A241716 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 - 2] == 2, AppendTo[A241716, t]], {n, 500}]; A241716
Select[Prime[Range[200]],PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Dec 09 2018 *)

A241732 Primes p such that p^3 + 2 and p^3 - 2 are semiprime.

Original entry on oeis.org

2, 11, 13, 17, 41, 89, 101, 239, 271, 331, 571, 641, 719, 1051, 1231, 1321, 1549, 1559, 1721, 1741, 1831, 1993, 1999, 2029, 2311, 2459, 2749, 2837, 2861, 2939, 3389, 3467, 3671, 4049, 4111, 4273, 4787, 4919, 4969, 5657, 5689, 5861, 6221, 6679, 6691, 6829, 7109

Offset: 1

K. D. Bajpai, Apr 27 2014

nonn

			11 is prime and appears in the sequence because 11^3 + 2 = 1333 = 31 * 43 and 11^3 - 2 = 1329 = 3 * 443, both are semiprime.
41 is prime and appears in the sequence because 41^3 + 2 = 68923 = 157 * 439 and 41^3 - 2 = 68919 = 3 * 22973, both are semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..2310

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

with(numtheory): KD:= proc() local k; k:=ithprime(n); if bigomega(k^3+2)=2 and bigomega(k^3-2)=2 then k; fi; end: seq(KD(), n=1..2000);

A241732 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2 && PrimeOmega[t^3 - 2] == 2, AppendTo[A241732, t]], {n, 500}]; A241732
Select[Prime[Range[1000]],PrimeOmega[#^3+2]==PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Jun 20 2019 *)

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

A241607 Semiprimes generated by the polynomial (1/4)(n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294*n - 6823316).

Original entry on oeis.org

5141923, 6084557, 11403823, 13201987, 17488411, 20017609, 33239291, 37446979, 42070423, 47139347, 72512623, 88747907, 118408673, 129881707, 169708339, 184952323, 201267887, 278376073, 324881567, 406044923, 436421497, 538566199, 616639427, 658920007, 750410069

Offset: 1

K. D. Bajpai, Apr 26 2014

nonn

(1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) is a well known prime producing polynomial found by Shyam Sunder Gupta, which generates 57 distinct primes for n = 0,1,...,55,56.

For n = 57, this polynomial yields the first semiprime: 5141923 = 821 * 6263.

			For n=57: (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) = 5141923 = 821 * 6263, which is a semiprime and is included in the sequence.
For n=58: (1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316) = 6084557 = 131 * 46447, which is a semiprime and is included in the sequence.

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

Cf. A007641 (for primes).

Cf. A001358, A072381, A082919, A121887, A145292, A228183, A237627.

with(numtheory): KD:= proc() local a,b,k; k:=(1/4)*(n^5 - 133*n^4 + 6729*n^3 - 158379*n^2 + 1720294*n - 6823316); a:=bigomega(k); if a=2 then RETURN (k); fi; end: seq(KD(), n=0..200);

A241607 = {}; Do[k= (1/4) * (n^5 - 133 * n^4 + 6729 * n^3 - 158379 * n^2 + 1720294 * n - 6823316); If[PrimeOmega[k] ==2, AppendTo[A241607, k]], {n,200}]; A241607
(*For the b-file:*) n=0;Do[t=((1/4) * (k^5 - 133 * k^4 + 6729 * k^3 - 158379 * k^2 + 1720294 * k - 6823316));If[PrimeOmega[t]==2, n++; Print[n," ",t]], {k,10^6}]

s=[]; for(n=1, 200, t=(1/4)*(n^5-133*n^4+6729*n^3-158379*n^2+1720294*n-6823316); if(bigomega(t)==2, s=concat(s, t))); s \\ Colin Barker, Apr 26 2014

A250292 Numbers k such that Pell(k) is a semiprime.

Original entry on oeis.org

7, 9, 17, 19, 23, 43, 47, 67, 73, 83, 103, 109, 139, 149, 157, 173, 179, 223, 239, 281, 311, 313, 349, 431, 557, 569, 577, 587

Offset: 1

Eric Chen, Dec 24 2014

a(29) >= 709. - Hugo Pfoertner, Jul 29 2019

859, 937, 1087, 1151, and 1193 belong to the sequence. 709 and 787 have not yet been ruled out. The next candidate after these appears to be 1471. - Daniel M. Jensen, Oct 18 2019

			17 is a term since Pell(17) = 1136689 = 137 * 8297 is a semiprime.

FactorDB, Status of Pell(709).
FactorDB, Status of Pell(787).

Cf. A001358, A096650, A072381, A085726, A000129.

pell:= gfun:-rectoproc({a(0)=0,a(1)=1,a(n)=2*a(n-1)+a(n-2)},a(n),remember):
filter:= proc(n) local F,f;
   F:= ifactors(pell(n),easy)[2];
   if add(f[2],f=F) > 2 then return false fi;
   if hastype(F,symbol) then
     if add(f[2],f=F) >= 2 then return false fi;
   else return evalb(add(f[2],f=F)=2)
   fi;
   F:= ifactors(pell(n))[2];
   evalb(add(f[2],f=F)=2)
end proc:
select(filter, [$1..230]); # Robert Israel, Jan 18 2016

a[0] = 0; a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + a[n - 2]; Select[Range[0, 160], PrimeOmega@ a@ # == 2 &] (* Michael De Vlieger, Jan 19 2016 *)

a(22)-a(23) from Daniel M. Jensen, Jan 18 2016
a(24) from Arkadiusz Wesolowski, Jan 19 2016
a(25)-a(27) from Sean A. Irvine, Jul 17 2017
a(28) from Sean A. Irvine, Jan 24 2018

cp's OEIS Frontend

A140987 Number of groups of order F(n).

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A241484 Primes p such that p+2 and p+4 are semiprime.

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Magma

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A241554 Semiprimes generated by the polynomial 2 * n^2 + 29.

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Maple

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PARI

A286545 Restricted growth sequence of A278245 (prime signature of Fibonacci numbers).

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A114812 Indices of Fibonacci numbers with 3 prime factors when counted with multiplicity.

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Mathematica

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A241716 Primes p such that p^3 - 2 is semiprime.

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Maple

Mathematica

A241732 Primes p such that p^3 + 2 and p^3 - 2 are semiprime.

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Maple

Mathematica

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

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A241607 Semiprimes generated by the polynomial (1/4)(n^5 - 133n^4 + 6729n^3 - 158379n^2 + 1720294*n - 6823316).