A079777 -id:A079777 - cp's OEIS frontend

A073853 Indices of zeros in A079777.

0, 5, 9, 12, 24, 45, 60, 65, 179, 764, 1268, 5891, 16135, 29909, 71774, 173310, 200040, 1454560, 2485272, 86430343, 92439810, 115854652, 7208007982, 17016737751, 17589706947, 24531053552, 33113576855, 80692537585, 234365843350, 266484243960, 285357252641, 426388494035, 975986718040, 1505420538689, 43633539697333

Offset: 1

Benoit Cloitre, Sep 02 2002

nonn

Let b(1) = b(2) = 1, b(k) = (b(k-1) + b(k-2)) mod k; sequence gives n such that b(n) = 0.
A079777(2^31-1) = 1103802855, and A079777(2^31) = 2117709557.
No further terms below k = 5*10^10, at which point, A079777(k-1) = 6059364906669 and A079777(k) = 29451014544130. - Luca Armstrong, Apr 07 2023

			b(3) = 2 mod 3 = 2; b(4) = (2+1) mod 4 = 3; b(5) = (3+2) mod 5 = 0, hence a(1) = 5.

Zak Seidov, A073853 Four more terms [From _Zak Seidov_, Dec 06 2009]

A079777(n) = 0.

class A073853 { public static void main(String [] args) { BigInteger an = BigInteger.ZERO ; BigInteger an1 = BigInteger.ONE ; BigInteger n = new BigInteger("2") ; for( ; ; n = n.add(BigInteger.ONE) ) { BigInteger an2 = an.add(an1).mod(n) ; if ( an2.compareTo(BigInteger.ZERO) == 0 ) System.out.println(n) ; an = an1 ; an1 = an2 ; } } } // R. J. Mathar, Dec 06 2009

a = 0; b = 1; lst = {0}; Do[c = Mod[a + b, n]; If[c == 0, AppendTo[lst, n]; Print@n]; a = b; b = c, {n, 2, 2^31}] (* Robert G. Wilson v *)

Corrected and extended by John W. Layman, Jun 11 2003
a(23)-a(26) from Zak Seidov; a(27)-a(28) from John W. Layman; a(29)-a(34) from Charles R Greathouse IV, Dec 09 2009. (These new terms were added by N. J. A. Sloane, Dec 20 2009.)
a(35) from Luca Armstrong, Apr 07 2023

A168360 n-b(n), where b(n) = A079777(n) = (b(n-1)+b(n-2) mod n); b(0)=0, b(1)=1.

Original entry on oeis.org

0, 0, 1, 1, 1, 5, 3, 4, 2, 9, 4, 5, 12, 7, 8, 3, 14, 3, 2, 8, 13, 3, 19, 2, 24, 4, 5, 12, 20, 6, 29, 7, 7, 17, 27, 12, 6, 21, 30, 15, 8, 26, 37, 23, 19, 45, 21, 22, 46, 22, 21, 46, 18, 14, 35, 52, 34, 32, 11, 46, 60, 48, 49, 37, 25, 65, 27, 28, 58, 20, 11, 34, 48, 12, 63, 3, 69, 75, 69, 68

Offset: 0

M. F. Hasler, Dec 05 2009

nonn

By definition, 0 <= A079777(n) < n, so this sequence satisfies 0 < a(n) <= n. The sequence A073853 gives the indices for which a(n)=n.

a=[0,n=1]; u=[1,1]~; print1("0, 0"); while(n<99, print1(", ",n++-a[1+bittest(n,0)]=a*u%n))

A168360 = A001477 - A079777

A096535 a(0) = a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 3, 5, 0, 5, 5, 10, 3, 0, 3, 3, 6, 9, 15, 5, 0, 5, 5, 10, 15, 0, 15, 15, 2, 17, 19, 5, 24, 29, 19, 13, 32, 8, 2, 10, 12, 22, 34, 13, 3, 16, 19, 35, 6, 41, 47, 37, 32, 16, 48, 9, 1, 10, 11, 21, 32, 53, 23, 13, 36, 49, 19, 1, 20, 21, 41, 62, 31, 20, 51, 71, 46, 40, 8, 48, 56

Offset: 0

Franklin T. Adams-Watters, Jun 23 2004

Suggested by Leroy Quet.
Three conjectures: (1) All numbers appear infinitely often, i.e., for every number k >= 0 and every frequency f > 0 there is an index i such that a(i) = k is the f-th occurrence of k in the sequence.
(2) a(j) = a(j-1) + a(j-2) and a(j) = a(j-1) + a(j-2) - j occur approximately equally often, i.e., lim_{n->infinity} x_n / y_n = 1, where x_n is the number of j <= n such that a(j) = a(j-1) + a(j-2) and y_n is the number of j <= n such that a(j) = a(j-1) + a(j-2) - j (cf. A122276).
(3) There are sections a(g+1), ..., a(g+k) of arbitrary length k such that a(g+h) = a(g+h-1) + a(g+h-2) for h = 1,...,k, i.e., the sequence is nondecreasing in these sections (cf. A122277, A122278, A122279). - Klaus Brockhaus, Aug 29 2006
a(A197877(n)) = n and a(m) <> n for m < A197877(n); see first conjecture. - Reinhard Zumkeller, Oct 19 2011

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A079777, A096274 (location of 0's), A096534, A132678.

a096535 n = a096535_list !! n
a096535_list = 1 : 1 : f 2 1 1 where
   f n x x' = y : f (n+1) y x where y = mod (x + x') n
-- Reinhard Zumkeller, Oct 19 2011

l = {1, 1}; For[i = 2, i <= 100, i++, len = Length[l]; l = Append[l, Mod[l[[len]] + l[[len - 1]], i]]]; l
f[s_] := f[s] = Append[s, Mod[s[[ -2]] + s[[ -1]], Length[s]]]; Nest[f, {1, 1}, 80] (* Robert G. Wilson v, Aug 29 2006 *)
RecurrenceTable[{a[0]==a[1]==1,a[n]==Mod[a[n-1]+a[n-2],n]},a,{n,90}] (* Harvey P. Dale, Apr 12 2013 *)

A104647 a(n) = a(n-1) mod n + a(n-2) mod n; a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 6, 6, 12, 8, 9, 17, 13, 16, 14, 14, 28, 24, 14, 18, 32, 28, 14, 18, 32, 24, 29, 25, 25, 50, 44, 30, 41, 37, 8, 9, 17, 26, 43, 29, 31, 60, 48, 20, 23, 43, 66, 61, 29, 40, 69, 57, 20, 23, 43, 66, 52, 60, 53, 53, 106, 97, 77, 46, 58, 104, 95, 63, 89, 82, 29, 39, 68

Offset: 0

Reinhard Zumkeller, Apr 21 2005

nonn

a(n) <= a(n-1) + a(n-2) < 2*n for n>0, see A105859 for numbers i such that a(i)=a(i-1)+a(i-2) and A105860 for numbers j with a(j)Reinhard Zumkeller, Apr 23 2005

A105855 = (first differences) and A105856 = (partial sums); records occur at A105857: A105858(n) = a(A105857(n)). - Reinhard Zumkeller, Apr 23 2005

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A079777

Fold[Append[#1, Mod[#1[[-1]], #2] + Mod[#1[[-2]], #2]] &, {0, 1}, Range[2, 73]] (* Ivan Neretin, Jun 18 2018 *)
nxt[{n_,a_,b_}]:={n+1,b,Mod[a,n+1]+Mod[b,n+1]}; NestList[nxt,{1,0,1},80][[;;,2]] (* Harvey P. Dale, May 12 2025 *)

A096534 a(1) = 0; a(2) = 1; a(n) = (a(n-1) + a(n-2)) mod n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 1, 6, 7, 3, 10, 1, 11, 12, 8, 4, 12, 16, 9, 5, 14, 19, 10, 5, 15, 20, 8, 0, 8, 8, 16, 24, 7, 31, 3, 34, 0, 34, 34, 28, 21, 7, 28, 35, 18, 7, 25, 32, 8, 40, 48, 36, 31, 13, 44, 1, 45, 46, 32, 18, 50, 6, 56, 62, 53, 49, 35, 16, 51, 67, 47, 42, 16, 58, 74, 56, 53, 31, 5, 36

Offset: 1

Franklin T. Adams-Watters, Jun 23 2004

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A079777, A096535.

A072987 is a closely related sequence.

N:= 100: # to get a(1)..a(N)
A:= Vector(N):
A[1]:= 0: A[2]:= 1:
for n from 3 to N do
  A[n]:= A[n-1] + A[n-2] mod n
od:
convert(A,list); # Robert Israel, Nov 13 2017

l = {0, 1}; For[i = 3, i <= 100, i++, len = Length[l]; l = Append[l, Mod[l[[len]] + l[[len - 1]], i]]]; l

A352406 The number of terms before reaching zero when starting at n and iterating: f(n) = n, f(n+1) = n+1; f(n+k) = (f(n+k-2) + f(n+k-1)) (mod (n+k)), where k>=2.

Original entry on oeis.org

5, 2, 7, 3, 8, 14, 5, 7, 4, 12, 46, 34, 18, 21, 21, 16, 5, 10, 8, 25, 128, 237, 79, 25, 266, 25, 10, 74, 34, 27, 6, 11, 22, 23, 72, 75, 26, 267, 16, 893, 28, 40, 8, 113, 27, 16, 163, 41, 13, 27, 169, 48, 837, 7, 88, 436, 23, 144, 59, 36, 77, 71, 466, 96, 14, 226, 371, 72, 231, 463, 377, 29

Offset: 0

Scott R. Shannon, Mar 15 2022

nonn

The first term corresponds to the number of terms before zero in A079777. This sequence starts at n and counts the number of terms before zero using the same iterative formula.

The sequence shows large variations in its value, e.g., a(496) = 222, a(497) = 1087851. In the first 50000 terms the largest value is a(21897) = 1248976431. In the same range the smallest number greater than 1 not to have appeared is 414, although it is likely all numbers eventually appear.

			a(0) = 5 as starting at 0 and 1 gives 0+1 % 2 = 1, 1+1 % 3 = 2, 1+2 % 4 = 3, 2+3 % 5 = 0, with five terms before reaching zero. See A079777.
a(1) = 2 as starting at 1 and 2 gives 1+2 % 3 = 0, with two terms before reaching zero. This is the smallest possible value and the only term to equal 2.
a(2) = 7 as starting at 2 and 3 gives 2+3 % 4 = 1, 3+1 % 5 = 4, 1+4 % 6 = 5, 4+5 % 7 = 2, 5+2 % 8 = 7, 2+7 % 9 = 0, with seven terms before reaching zero.

Scott R. Shannon, Line graph of the first 50000 terms.

Cf. A079777, A352407 (multiplication).

A333453 Binary concatenation (ignoring leading zeros) of a(n-1) and a(n-2) mod n, starting with a(n) = n for n <= 1.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 3, 0, 3, 3, 5, 1, 1, 3, 7, 1, 15, 14, 5, 18, 9, 12, 3, 14, 11, 15, 17, 17, 1, 20, 11, 0, 11, 11, 17, 3, 5, 23, 37, 37, 5, 29, 27, 33, 27, 6, 35, 4, 3, 28, 15, 49, 19, 46, 33, 13, 25, 14, 9, 40, 49, 4, 57, 19, 57, 23, 11, 40, 39, 52, 7, 3, 31

Offset: 0

Alois P. Heinz, Mar 21 2020

Value 0 is treated as empty bit string.

			a(18) = 5 = 239 mod 18, where 239 = 11101111_2 is the binary concatenation 1110_2 = 14 = a(17) and 1111_2 = 15 = a(16).

Alois P. Heinz, Table of n, a(n) for n = 0..32768

Cf. A062383, A063896, A079777.

a:= proc(n) option remember; `if`(n<2, n, (t-> a(n-1)*
      `if`(t=0, 1, 2^(ilog2(t)+1))+t)(a(n-2)) mod n)
    end:
seq(a(n), n=0..100);

a(n) = (a(n-1)*A062383(a(n-2)) + a(n-2)) mod n if n > 1, a(n) = n if n < 2.

A352407 The number of terms before reaching zero when starting at n and iterating: f(n) = n, f(n+1) = n+1; f(n+k) = (f(n+k-2) * f(n+k-1)) (mod (n+k)), where k>=2.

Original entry on oeis.org

2, 3, 14, 5, 5, 11, 4, 42, 54, 6, 17, 38, 6, 27, 12, 71, 20, 5, 6, 8, 12, 12, 42, 37, 36, 23, 22, 9, 5, 19, 10, 35, 31, 31, 60, 47, 33, 44, 46, 15, 8, 49, 14, 9, 12, 23, 35, 34, 28, 11, 86, 43, 20, 49, 18, 17, 12, 9, 22, 45, 26, 5, 31, 51, 72, 7, 6, 121, 120, 111, 86, 341, 56, 63, 12, 85, 12, 21

Offset: 0

Scott R. Shannon, Mar 15 2022

nonn

This sequences uses the same iterative formula as A352406 except that the two previous terms are multiplied instead of added. See that sequence for further details.

In the first 500000 terms the largest value is a(409758) = 1480452. In the same range the smallest number greater than 1 not to have appeared is 16291, although it is likely all numbers eventually appear.

			a(0) = 2 as starting at 0 and 1 gives 0*1 % 2 = 0, with two terms before reaching zero. This is the smallest possible value and the only term to equal 2.
a(2) = 14 as starting at 2 and 3 gives 2*3 % 4 = 2, 3*2 % 5 = 1, 2*1 % 6 = 2, 1*2 % 7 = 2, 2*2 % 8 = 4, 2*4 % 9 = 8, 4*8 % 10 = 2, 8*2 % 11 = 5, 2*5 % 12 = 10, 5*10 % 13 = 11, 10*11 % 14 = 12, 11*12 % 15 = 12, 12*12 % 16 = 0, with fourteen terms before reaching zero.
a(3) = 5 as starting at 3 and 4 gives 3*4 % 5 = 2, 4*2 % 6 = 2, 2*2 % 7 = 4, 2*4 % 8 = 0, with five terms before reaching zero.

Cf. A352406 (addition), A079777.

cp's OEIS Frontend

A073853 Indices of zeros in A079777.

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A168360 n-b(n), where b(n) = A079777(n) = (b(n-1)+b(n-2) mod n); b(0)=0, b(1)=1.

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A096535 a(0) = a(1) = 1; a(n) = (a(n-1) + a(n-2)) mod n.

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A104647 a(n) = a(n-1) mod n + a(n-2) mod n; a(0) = 0, a(1) = 1.

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A096534 a(1) = 0; a(2) = 1; a(n) = (a(n-1) + a(n-2)) mod n.

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A352406 The number of terms before reaching zero when starting at n and iterating: f(n) = n, f(n+1) = n+1; f(n+k) = (f(n+k-2) + f(n+k-1)) (mod (n+k)), where k>=2.

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A333453 Binary concatenation (ignoring leading zeros) of a(n-1) and a(n-2) mod n, starting with a(n) = n for n <= 1.

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A352407 The number of terms before reaching zero when starting at n and iterating: f(n) = n, f(n+1) = n+1; f(n+k) = (f(n+k-2) * f(n+k-1)) (mod (n+k)), where k>=2.

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