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1
2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70
COMMENTS
Primes are a subsequence, but semiprimes (cf. A001358) are not: first missing: A249406(1)=6=2*3, A249406(732)=2196323=1481*1483, A249406(1263)=6502499=2549*2551, A249406(1472)=8820899=2969*2971, A249406(1756)=12531599=3539*3541, ... ; see A249411 for numbers of the form m*(m+1).
PROG
(Haskell)
import Data.List ((\\))
a249407 n = a249407_list !! (n-1)
a249407_list = f [2..] where
f ws@(u:v:_) = u : v : f (ws \\ [u, v, u * v])
Numbers in A249406 that are not of the form m*(m+1).
+20
2
1, 399, 8099, 33123, 93635, 159200, 256035, 492803, 864899, 1416099, 2196323, 3261635, 4674243, 6502499, 8820899, 12531599, 16257023, 20757135, 26132543, 32489999, 39942399, 48608783, 58614335, 65593800, 73205135, 86713343, 102009999, 119246399, 138579983
PROG
(Haskell)
import Data.List ((\\))
a249408 n = a249408_list !! (n-1)
a249408_list = filter ((== 0) . a005369) a249406_list
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.
+10
22
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
COMMENTS
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
FORMULA
See Zaslavsky (2016) link.
MAPLE
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))];
MATHEMATICA
(* 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}];
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}}]
********
(* 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]]
********
(* 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]]
PROG
(Haskell)
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
(Python)
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
(Python)
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
2, 3, 6, 18, 108, 1944, 209952, 408146688, 85691213438976, 34974584955819144511488, 2997014624388697307377363936018956288, 104819342594514896999066634490728502944926883876041385836544
MATHEMATICA
nxt[{a_, b_}]:={b, a*b}; Transpose[NestList[nxt, {2, 3}, 12]][[1]] (* Harvey P. Dale, Nov 16 2014 *)
PROG
(Haskell)
a000304 n = a000304_list !! (n-2)
a000304_list = 2 : 3 : zipWith (*) a000304_list (tail a000304_list)
2, 12, 30, 42, 72, 110, 156, 210, 272, 342, 380, 420, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832
EXAMPLE
Distribution of oblong numbers on A249406 and its complement, m*(m+1) factorizations are shown in respective columns:
. ---+---------+---------+-------- ---+---------+---------+--------
. 1 | 2 | | 1*2 13 | 182 | 13*14 |
. 2 | 6 | 2*3 | 14 | 210 | | 14*15
. 3 | 12 | | 3*4 15 | 240 | 15*16 |
. 4 | 20 | 4*5 | 16 | 272 | | 16*17
. 5 | 30 | | 5*6 17 | 306 | 17*18 |
. 6 | 42 | | 6*7 18 | 342 | | 18*19
. 7 | 56 | 7*8 | 19 | 380 | | 19*20
. 8 | 72 | | 8*9 20 | 420 | | 20*21
. 9 | 90 | 9*10 | 21 | 462 | | 21*22
. 10 | 110 | | 10*11 22 | 506 | 22*23 |
. 11 | 132 | 11*12 | 23 | 552 | | 23*24
. 12 | 156 | | 12*13 24 | 600 | 24*25 | .
PROG
(Haskell)
a249411 n = a249411_list !! (n-1)
a249411_list = filter ((== 1) . a005369) a249407_list
a(1)=0; the next term is always the product of the two smallest numbers not yet in the sequence and which have not yet been used.
+10
1
0, 2, 12, 30, 56, 90, 143, 210, 272, 342, 420, 506, 600, 702, 812, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3135, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8372, 8742, 9120, 9506, 9900
EXAMPLE
Start with a(1) = 0. The missing numbers are 1 2 3 4 5 6 ...
Multiply the first two, and we get 2, which is therefore a(2).
Cross 1, 2, and 1*2 = 2 off the missing list.
The first two missing numbers are now 3 and 4, so a(3) = 3*4 = 12.
Cross off 3,4,12 from the missing list.
Repeat!
MAPLE
M:=50; A:=[0]; miss:=[seq(n, n=1..M^2)]:
for n from 1 to M do t1:=miss[1]*miss[2]; A:=[op(A), t1];
miss:=[seq(miss[i], i=3..nops(miss))];
miss:=remove('x->x=t1', miss);
od:
A;
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