A105755 -id:A105755 - cp's OEIS frontend

A127208 Union of all n-step Lucas sequences, that is, all sequences s(1-n) = s(2-n) = ... = s(-1) = -1, s(0) = n and for k > 0, s(k) = s(k-1) + ... + s(k-n).

1, 3, 4, 7, 11, 15, 18, 21, 26, 29, 31, 39, 47, 51, 57, 63, 71, 76, 99, 113, 120, 123, 127, 131, 191, 199, 223, 239, 241, 247, 255, 322, 367, 439, 443, 475, 493, 502, 511, 521, 708, 815, 843, 863, 943, 983, 1003, 1013, 1023, 1364, 1365, 1499, 1695, 1871, 1959

Offset: 1

T. D. Noe, Jan 09 2007

nonn

Noe and Post conjectured that the only positive terms that are common to any two distinct n-step Lucas sequences are the Mersenne numbers (A001348) that begin each sequence and 7 and 11 (in 2- and 3-step) and 5071 (in 3- and 4-step). The intersection of this sequence with the union of all the n-step Fibonacci sequences (A124168) appears to consist of 4, 21, 29, the Mersenne numbers 2^n-1 for all n and the infinite set of Eulerian numbers in A127232.

T. D. Noe, Table of n, a(n) for n=1..1000
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A227885.

LucasSequence[n_,kMax_] := Module[{a,s,lst={}}, a=Join[Table[ -1,{n-1}],{n}]; While[s=Plus@@a; a=RotateLeft[a]; a[[n]]=s; s<=kMax, AppendTo[lst,s]]; lst]; nn=10; t={}; Do[t=Union[t,LucasSequence[n,2^(nn+1)]], {n,2,nn}]; t

Union(A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755,...)

A125129 Partial sums of diagonals of array of k-step Lucas numbers as in A125127, read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 8, 11, 1, 12, 19, 26, 1, 19, 33, 45, 57, 1, 30, 58, 84, 102, 120, 1, 48, 101, 157, 197, 222, 247, 1, 77, 179, 292, 380, 436, 469, 502, 1, 124, 318, 546, 731, 855, 929, 971, 1013, 1, 200, 567, 1026, 1409, 1674, 1838, 1932, 1984, 2036

Offset: 1

Jonathan Vos Post, Nov 23 2006

Array of partial sums of diagonals of L(k,n) begins: 0.|.1...4..11...26...57..120..247..502.1013.2036.
|.1...8..19...45..102..222..469..971.1984.
|.1..12..33...84..197..436..929.1932.
|.1..19..58..157..380..855.1838.
|.1..30.101..292..731.1674.
|.1..48.179..546.1409.
|.1..77.318.1026.
|.1.124.567.
|.1.200.
|.1.

			Row 1 of the derived array is the partial sum of the diagonal above the main diagonal of array of k-step Lucas numbers as in A125127, hence the partial sums of: 1, 7, 11, 26, 57, 120, 247, 502, 103, ... are 1 = 1; 8 = 1 + 7; 19 = 1 + 7 + 11; 45 = 1 + 7 + 11 + 26; and so forth.

Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125127, A000295.

Row 0 = SUM[i=1..n]L(i,i) = A127128 = partial sum of main diagonal of array of A125127. Row 1 = SUM[i=1..n]L(i,i+1) = partial sum of diagonal above main diagonal of array of A125127. Row 2 = SUM[i=1..n]L(i,i+2) = partial sum of diagonal 2 above main diagonal of array of A125127. .. Row m = SUM[i=1..n]L(i,i+m) = partial sum of diagonal 2 above main diagonal of array of A125127.

A227885 Primes in the union of all n-step Lucas sequences.

Original entry on oeis.org

2, 3, 7, 11, 29, 31, 47, 71, 113, 127, 131, 191, 199, 223, 239, 241, 367, 439, 443, 521, 863, 983, 1013, 1499, 1871, 2003, 2207, 3571, 6553, 8087, 8191, 9349, 16369, 32647, 32707, 36319, 63487, 65407, 65519, 122401, 126719, 131071, 196331, 260111, 524287

Offset: 1

Robert Price, Oct 25 2013

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000 (first 383 terms from Robert Price)
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A127208, A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755.

plst={2}; plimit=10^39; For[n=2, n<=3+Log[2,plimit], n++, llst={}; For[i=1, i

2 and the primes in A127208.

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A127208 Union of all n-step Lucas sequences, that is, all sequences s(1-n) = s(2-n) = ... = s(-1) = -1, s(0) = n and for k > 0, s(k) = s(k-1) + ... + s(k-n).

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A125129 Partial sums of diagonals of array of k-step Lucas numbers as in A125127, read by antidiagonals.

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A227885 Primes in the union of all n-step Lucas sequences.

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