A075327 -id:A075327 - cp's OEIS frontend

A075326 Anti-Fibonacci numbers: start with a(0) = 0, and extend by the rule that the next term is the sum of the two smallest numbers that are not in the sequence nor were used to form an earlier sum.

0, 3, 9, 13, 18, 23, 29, 33, 39, 43, 49, 53, 58, 63, 69, 73, 78, 83, 89, 93, 98, 103, 109, 113, 119, 123, 129, 133, 138, 143, 149, 153, 159, 163, 169, 173, 178, 183, 189, 193, 199, 203, 209, 213, 218, 223, 229, 233, 238, 243, 249, 253, 258, 263, 269, 273, 279, 283

Offset: 0

Amarnath Murthy, Sep 16 2002

In more detail, the sequence is constructed as follows: Start with a(0) = 0. The missing numbers are 1 2 3 4 5 6 ... Add the first two, and we get 3, which is therefore a(1). Cross 1, 2, and 1+2=3 off the missing list. The first two missing numbers are now 4 and 5, so a(2) = 4+5 = 9. Cross off 4,5,9 from the missing list. Repeat.
In other words, this is the sum of consecutive pairs in the sequence 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, ..., (A249031) the complement to the present one in the natural numbers. For example, a(1)=1+2=3, a(2)=4+5=9, a(3)=6+7=13, ... - Philippe Lallouet (philip.lallouet(AT)orange.fr), May 08 2008
The new definition is due to Philippe Lalloue (philip.lallouet(AT)orange.fr), May 08 2008, while the name "anti-Fibonacci numbers" is due to D. R. Hofstadter, Oct 23 2014.
Original definition: second members of pairs in A075325.
If instead we take the sum of the last used non-term and the most recent (i.e., 1+2, 2+4, 4+5, 5+7, etc.), we get A008585. - Jon Perry, Nov 01 2014
The sequences a = A075325, b = A047215, and c = A075326 are the solutions of the system of complementary equations defined recursively as follows:
  a(n) = least new,
  b(n) = least new,
  c(n) = a(n) + b(n),
where "least new k" means the least positive integer not yet placed. For anti-tribonacci numbers, see A265389; for anti-tetranacci, see A299405. - Clark Kimberling, May 01 2018
We see the Fibonacci numbers 3, 13, 89 and 233 occur in this sequence of anti-Fibonacci numbers. Are there infinitely many Fibonacci numbers occurring in (a(n))? The answer is yes: at least 13% of the Fibonacci numbers occur in (a(n)). This follows from Thomas Zaslavsky's formula, which implies that the sequence A017305 = (10n+3) is a subsequence of (a(n)). The Fibonacci sequence A000045 modulo 10 equals A003893, and has period 60. In this period, the number 3 occurs 8 times. - Michel Dekking, Feb 14 2019
From Augusto Santi, Aug 16 2025: (Start)
If we apply the anti-Fibonacci algorithm to the set of natural numbers minus the multiples of 3, we get 5, 10, 20, 25, 35, 40, 50, ...; that is, all the multiples of 5 present in the restricted set used. It is quite curious that in this particular case the algorithm can be applied recursively to its own output, generating, at the generic step s, the subset of multiples of 5^s (see Mathematics StackExchange link).
Conjectures:
After the first 0, the residues (mod 5) all fall in the classes 3 and 4. More generally, for k-nacci sequences the residue classes (mod k^2+1) all fall in k consecutive ones, the first being ceiling((k^2+1)/2​).
It is known that the sequence contains the arithmetic progression 10k+3, 20k+9 and 40k+18. These three progressions cover, experimentally, the 87.5% = 7/8 of the entire sequence. The remaining terms all belong to two forms: 40k+38 and 40k+39.
The anti-Fibonacci sequence contains all the squares of the numbers of the form 10k+3 and 10k+7, and all the cubes of the numbers of the form 10k+7, for k>=0. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Wieb Bosma, Rene Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, and Thian Tromp, Using Walnut to solve problems from the OEIS, arXiv:2503.04122 [math.NT], 2025. See pp. 7, 14.
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 18.
D. R. Hofstadter, Anti-Fibonacci numbers, Oct 23 2014.
Augusto Santi, A conjecture on Anti-k-nacci numbers, Mathematics StackExchange, 2025.
Thomas Zaslavsky, Anti-Fibonacci Numbers: A Formula, Sep 26 2016

Cf. A008585, A075325, A075327, A249031, A249032 (first differences), A000045.

Cf. also A079523, A131323, A249031, A249032, A249406.

import Data.List ((\\))
a075326 n = a075326_list !! n
a075326_list = 0 : f [1..] where
   f ws@(u:v:_) = y : f (ws \\ [u, v, y]) where y = u + v
-- Reinhard Zumkeller, Oct 26 2014

# Maple code for M+1 terms of sequence, from N. J. A. Sloane, Oct 26 2014
c:=0; a:=[c]; t:=0; M:=100;
for n from 1 to M do
s:=t+1; if s in a then s:=s+1; fi;
t:=s+1; if t in a then t:=t+1; fi;
c:=s+t;
a:=[op(a),c];
od:
[seq(a[n],n=1..nops(a))];

(* Three sequences a,b,c as in Comments *)
z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {};
Do[AppendTo[a,
   mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];
  AppendTo[c, Last[a] + Last[b]], {z}];
Take[a, 100] (* A075425 *)
Take[b, 100] (* A047215 *)
Take[c, 100] (* A075326 *)
Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
  Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
Alignment -> ".",
Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}]
(* Peter J. C. Moses, Apr 26 2018 *)
********
(* Sequence "a" via A035263 substitutions *)
Accumulate[Prepend[Flatten[Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 0}}] &, {0}, 7] /. Thread[{0, 1} -> {{5, 5}, {6, 4}}]], 3]]
(* Peter J. C. Moses, May 01 2018 *)
********
(* Sequence "a" via Hofstadter substitutions; see his 2014 link *)
morph = Rest[Nest[Flatten[#/.{1->{3},3->{1,1,3}}]&,{1},6]]
hoff = Accumulate[Prepend[Flatten[morph/.Thread[{1,3}->{{6,4,5,5},{6,4,6,4,6,4,5,5}}]],3]]
(* Peter J. C. Moses, May 01 2018 *)

def aupton(nn):
    alst, disallowed, mink = [0], {0}, 1
    for n in range(1, nn+1):
        nextk = mink + 1
        while nextk in disallowed: nextk += 1
        an = mink + nextk
        alst.append(an)
        disallowed.update([mink, nextk, an])
        mink = nextk + 1
        while mink in disallowed: mink += 1
    return alst
print(aupton(57)) # Michael S. Branicky, Jan 31 2022

def A075326(n): return 5*n-1-int((n|(~((m:=n-1>>1)+1)&m).bit_length())&1) if n else 0 # Chai Wah Wu, Sep 11 2024

See Zaslavsky (2016) link.

More terms from David Wasserman, Jan 16 2005

Entry revised (including the addition of an initial 0) by N. J. A. Sloane, Oct 26 2014 and Sep 26 2016 (following a suggestion from Thomas Zaslavsky)

A075325 Pair the natural numbers such that the m-th pair is (r, s) where r, s and s-r are the smallest numbers which have not occurred earlier and also are not equal to the difference of any earlier pair: (1, 3), (4, 9), (6, 13), (8, 18), (11, 23), (14, 29), (16, 33), (19, 39), (21, 43), (24, 49), (26, 53), (28, 58), ... Sequence gives first term of each pair.

Original entry on oeis.org

1, 4, 6, 8, 11, 14, 16, 19, 21, 24, 26, 28, 31, 34, 36, 38, 41, 44, 46, 48, 51, 54, 56, 59, 61, 64, 66, 68, 71, 74, 76, 79, 81, 84, 86, 88, 91, 94, 96, 99, 101, 104, 106, 108, 111, 114, 116, 118, 121, 124, 126, 128, 131, 134, 136, 139, 141, 144, 146, 148, 151, 154, 156

Offset: 1

Amarnath Murthy, Sep 16 2002

nonn

Most of the pairs are of the form (r,2r+1) except for the ones like a(4) = (8,18) and a(12) = (28,58) and (38,78) etc. which are of the form (r,2r +2).

			The first pair (1, 3) covers 1, 2, 3. The second pair is (4, 9) covering 4, 5, 9.

Clark Kimberling, Table of n, a(n) for n = 1..10000
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See p. 4.

Cf. A007814, A075326, A075327.

The sequence formed by listing the differences between the second and first elements of each pair is A047215.

(* Here, the offset for (a(n)) is 0. *)
z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {};
Do[AppendTo[a,
   mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];
  AppendTo[c, Last[a] + Last[b]], {z}];
Take[a, 100] (* A075325 *)
Take[b, 100] (* A047215 *)
Take[c, 100] (* A075326 *)
Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
  Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
 Alignment -> ".",
 Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}]
(* Peter J. C. Moses, Apr 26 2018 *)

used = vector(500); i = 1; A = vector(80); B = A; C = A; for (n = 1, 80, while (used[i], i++); j = i + 1; while (used[j] || used [i + j], j++); A[n] = i; B[n] = i + j; C[n] = i + i + j; used[i] = 1; used[j] = 1; used[i + j] = 1); A \\ David Wasserman, Jan 16 2005

Let A(n) = A007814(n). Let B(n) = A(n) + 1 if A(n) < 2; B(n) = 0 if A(n)>=2 & A(n) is even; B(n) = 2 if A(n) >= 2 & A(n) is odd. Then a(n) = (5n+B(n)-4)/2. - John Chew (jjchew(AT)math.utoronto.ca), Jun 20 2006

More terms from David Wasserman, Jan 16 2005

A035166 Let d(m) = denominator of Sum_{k=1..m} 1/k^2 and consider f(m) = product of primes which appear to odd powers in d(m); sequence lists m such that f(m) is different from f(m-1).

Original entry on oeis.org

1, 10, 15, 20, 25, 42, 49, 50, 55, 66, 75, 78, 91, 100, 110, 121, 125, 136, 153, 156, 164, 169, 171, 182, 189, 190, 205, 250, 253, 272, 276, 289, 294, 342, 343, 354, 361, 375, 406, 413, 435, 465, 473, 496, 500, 506, 516, 529, 555, 592, 605, 625

Offset: 1

Bill Gosper, Sep 04 2002

The prime 479 first appears in f(m) at m = 2395, ahead of 71, which first appears in f(2485).
The first occurrence of four distinct primes is at m = 2500, with 5^7, 17^3, 71 and 479.
For 1890 < m < 2006, d(m) is a square (f(m)=1). The lone prime in 1875 .. 1890 is 61 and in 2006 .. 2027 it is 59.
It appears that f(m) can differ from f(m-1) in at most one prime.
(f from definition) = A007913, squarefree part. - Reinhard Zumkeller, Jul 06 2012

			f(10) = 5 is the first time f(m) > 1. The 5 persists until it disappears at m = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A075326, A075327, A007407.

import Data.List (findIndices)
a035166 n = a035166_list !! (n-1)
a035166_list = map (+ 1) $ findIndices (/= 0) $ zipWith (-) (tail gs) gs
   where gs = 0 : map a007913 a007407_list
-- Reinhard Zumkeller, Jul 06 2012

for k:1 do (subset(factor_number(denom(harmonic(k,2))), lambda([x],oddp(second(x)))), if old#old:%% then print(k,%%))

d[n_] := Denominator[ HarmonicNumber[n, 2]]; f[n_] := Times @@ Select[ FactorInteger[d[n]], OddQ[#[[2]]]&][[All, 1]]; A035166 = Join[{1}, Select[ Range[1000], f[#] != f[#-1]&]] (* Jean-François Alcover, Feb 26 2016 *)

d(m) = denominator(sum(k=1, m, 1/k^2));
f(m) = my(f=factor(d(m))); for (k=1, #f~, if (!(f[k,2] % 2), f[k,2] = 0)); factorback(f);
isok(m) = if (m==1, 1, f(m) != f(m-1)); \\ Michel Marcus, Sep 06 2023

cp's OEIS Frontend

A075326 Anti-Fibonacci numbers: start with a(0) = 0, and extend by the rule that the next term is the sum of the two smallest numbers that are not in the sequence nor were used to form an earlier sum.

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A035166 Let d(m) = denominator of Sum_{k=1..m} 1/k^2 and consider f(m) = product of primes which appear to odd powers in d(m); sequence lists m such that f(m) is different from f(m-1).

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A075328 Difference between n-th pair in A075325.

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