A257227
Indices of primes in the 10th-order Fibonacci number sequence, A122265.
Original entry on oeis.org
11, 361, 373, 2440, 14002, 68990
Offset: 1
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a={0, 0, 0, 0, 0, 0, 0, 0, 0, 1}; step=10; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst
Flatten[Position[LinearRecurrence[{1,1,1,1,1,1,1,1,1,1},{0,0,0,0,0,0,0,0,0,1},69000],?PrimeQ]]-1 (* _Harvey P. Dale, Dec 08 2017 *)
A257228
Primes in the 10th-order Fibonacci numbers A122265.
Original entry on oeis.org
2, 3876345660966505581780035851822613413637045687942554538584103751904155528656612320450718024564637501177857, 15784273697726525594915158437704910106795669967932151790483411869827615323130147795459165734845011296559523773
Offset: 1
-
a={0, 0, 0, 0, 0, 0, 0, 0, 0, 1}; step=10; offset=0; lst={}; For[n=step+offset,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst
A251759
10-step Fibonacci sequence starting with 0,0,0,0,0,0,0,0,1,0.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1022, 2043, 4084, 8164, 16320, 32624, 65216, 130368, 260608, 520960, 1041409, 2081796, 4161549, 8319014, 16629864, 33243408, 66454192, 132843168, 265555968, 530851328, 1061181696, 2121321983
Offset: 0
- Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1).
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LinearRecurrence[Table[1, {10}], {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, 45] (* Michael De Vlieger, Dec 08 2014 *)
A251760
10-step Fibonacci sequence starting with 0,0,0,0,0,0,0,1,0,0.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 4, 8, 16, 32, 64, 128, 255, 510, 1020, 2039, 4076, 8148, 16288, 32560, 65088, 130112, 260096, 519937, 1039364, 2077708, 4153377, 8302678, 16597208, 33178128, 66323696, 132582304, 265034496, 529808896, 1059097855, 2117156346
Offset: 0
- Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1).
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LinearRecurrence[Table[1, {10}], {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, 45] (* Michael De Vlieger, Dec 08 2014 *)
A105758
Indices of prime hexanacci (or Fibonacci 6-step) numbers A001592 (using offset -4).
Original entry on oeis.org
3, 36, 37, 92, 660, 6091, 8415, 11467, 13686, 38831, 49828, 97148
Offset: 1
Cf.
A000045,
A000073,
A000078 (and
A001631),
A001591,
A122189 (or
A066178),
A079262,
A104144,
A122265,
A168082,
A168083 (Fibonacci, tribonacci, tetranacci numbers and other generalizations).
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a={1, 0, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, n]], {n, 30000}]; lst
A302990
a(n) = index of first odd prime number in the (n-th)-order Fibonacci sequence Fn, or 0 if no such index exists.
Original entry on oeis.org
0, 0, 4, 6, 9, 10, 40, 14, 17, 19, 361, 23, 90, 26, 373, 47, 288, 34, 75, 38, 251, 43, 67, 47, 74, 310, 511, 151534, 57, 20608, 1146, 62, 197, 94246, 9974, 287, 271172, 758
Offset: 0
a(2) = 4 because F2 (Fibonacci) = 0, 1, 1, 2, 3, 5, 8, ... and F2(4) = 3 is prime.
a(3) = 6 because F3 (tribonacci) = 0, 0, 1, 1, 2, 4, 7, 13, ... and F3(6) = 7 is prime.
a(4) = 9 because F4 (tetranacci) = 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, ... and F4(9) = 29 is prime.
From _M. F. Hasler_, Apr 18 2018: (Start)
We see that Fn(k) = 2^(k-n) for n <= k < 2n and thus Fn(2n) = 2^n-1, so a(n) = 2n exactly for the Mersenne prime exponents A000043.
a(n) = 2n + 1 when 2^(n+1) - 3 is prime (n+1 in A050414) but 2^n-1 is not, i.e., n = 4, 8, 9, 11, 21, 23, 28, 93, 115, 121, 149, 173, 212, 220, 232, 265, 335, 451, 544, 688, 693, 849, 1735, ...
For other primes we have: a(29) = 687*30 - 2, a(37) = 20*38 - 2, a(41) > 10^4, a(43) > 10^4, a(47) = 5*48 - 1, a(53) = 53*54 - 2, a(59) = 307*60 - 2, a(67) = 6*67 - 1. (End)
Cf.
A001605 (indices of prime numbers in F2).
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A302990(n,L=oo,a=vector(n+1,i,if(i1 && for(i=-2+2*n+=1,L, ispseudoprime(a[i%n+1]=2*a[(i-1)%n+1]-a[i%n+1]) && return(i))} \\ Testing primality only for i%n>n-3 is not faster, even for large n. - M. F. Hasler, Apr 17 2018; improved Apr 18 2018
Original entry on oeis.org
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2047, 4092, 8180, 16352, 32688, 65344, 130624, 261120, 521984, 1043456, 2085888, 4169729, 8335366, 16662552, 33308752, 66584816, 133104288, 266077952, 531894784, 1063267584
Offset: 0
a(12)=C(12,12)*2^12-C(2,1)*2^1=4092.
- Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,0,0,0,0,0,-1).
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k:=10:taylor(1/(1-2*z+z^(k+1)),z=0,30); for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od;
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