A374356 -id:A374356 - cp's OEIS frontend

A377355 a(n) is the greatest k not yet in the sequence such that A374356(n) = A374356(k).

0, 1, 3, 2, 6, 7, 4, 5, 12, 13, 15, 14, 8, 9, 11, 10, 24, 25, 27, 26, 30, 31, 28, 29, 16, 17, 19, 18, 22, 23, 20, 21, 48, 49, 51, 50, 54, 55, 52, 53, 60, 61, 63, 62, 56, 57, 59, 58, 32, 33, 35, 34, 38, 39, 36, 37, 44, 45, 47, 46, 40, 41, 43, 42, 96, 97, 99, 98

Offset: 0

Rémy Sigrist, Oct 27 2024

This sequence is a self-inverse permutation of the nonnegative integers that preserves the binary length.

			The first terms are:
  n   a(n)  A374356(n)  A374356(a(n))
  --  ----  ----------  -------------
   0     0           0              0
   1     1           1              1
   2     3           2              2
   3     2           2              2
   4     6           4              4
   5     7           5              5
   6     4           4              4
   7     5           5              5
   8    12           8              8
   9    13           9              9
  10    15          10             10
  11    14          10             10
  12     8           8              8
  13     9           9              9
  14    11          10             10
  15    10          10             10

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A374356.

PARI
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\\ See Links section.
```

A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 6, 5, 8, 2, 4, 4, 6, 4, 8, 6, 10, 3, 6, 6, 9, 5, 10, 8, 13, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 3, 6, 6, 9, 6, 12, 9, 15, 5, 10, 10, 15, 8, 16, 13, 21, 2, 4, 4, 6, 4, 8, 6, 10, 4, 8, 8, 12, 6, 12, 10, 16, 4, 8, 8, 12, 8, 16, 12, 20, 6, 12, 12, 18

Offset: 0

N. J. A. Sloane, Aug 10 2014; revised Sep 05 2014

This is the Run Length Transform of S(n) = Fibonacci(n+2).
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).
Also the number of sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference. The maximal case is A384883. For prime instead of binary indices we have A166469. - Gus Wiseman, Jul 05 2025

			From _Gus Wiseman_, Jul 05 2025: (Start)
The binary indices of 11 are {1,2,4}, with sparse subsets {{},{1},{2},{4},{1,4},{2,4}}, so a(11) = 6.
The maximal runs of binary indices of 11 are ((1,2),(4)), with lengths (2,1), so a(11) = F(2+2)*F(1+2) = 6.
The a(0) = 1 through a(12) = 3 sparse subsets are:
  0    1    2    3    4    5    6    7    8    9    10    11    12
  ------------------------------------------------------------------
  {}   {}   {}   {}   {}   {}   {}   {}   {}   {}    {}    {}    {}
       {1}  {2}  {1}  {3}  {1}  {2}  {1}  {4}  {1}   {2}   {1}   {3}
                 {2}       {3}  {3}  {2}       {4}   {4}   {2}   {4}
                           {1,3}     {3}       {1,4} {2,4} {4}
                                     {1,3}                 {1,4}
                                                           {2,4}
The greatest number whose set of binary indices is a member of column n above is A374356(n).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Cf. A245562, A000045, A001045, A071053, A245565, A246028.
A034839 counts subsets by number of maximal runs, strict partitions A116674.
A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000071, A001629, A010049, A053538, A166469, A202023, A202064, A208342, A374356, A384883.

with(combinat); ans:=[];
for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
   elif out1 = 0 and t1[i] = 1 then c:=c+1;
   elif out1 = 1 and t1[i] = 0 then c:=c;
   elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;
   fi;
   if i = L1 and c>0 then lis:=[c,op(lis)]; fi;
                   od:
a:=mul(fibonacci(i+2), i in lis);
ans:=[op(ans),a];
od:
ans;

a[n_] := Sum[Mod[Binomial[3k, k] Binomial[n, k], 2], {k, 0, n}];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 29 2020, after Chai Wah Wu *)
spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[spars[bpe[n]]],{n,0,30}] (* Gus Wiseman, Jul 05 2025 *)

a(n)=my(s=1,k); while(n, n>>=valuation(n,2); k=valuation(n+1,2); s*=fibonacci(k+2); n>>=k); s \\ Charles R Greathouse IV, Oct 21 2016

# use RLT function from A278159
from sympy import fibonacci
def A245564(n): return RLT(n,lambda m: fibonacci(m+2)) # Chai Wah Wu, Feb 04 2022

a(n) = Sum_{k=0..n} ({binomial(3k,k)*binomial(n,k)} mod 2). - Chai Wah Wu, Oct 19 2016

A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 4, 5, 2, 4, 2, 5, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 1, 2, 9, 10, 4, 8, 4, 5, 8, 9, 4, 10, 5, 10, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, 1, 2, 17, 18, 0, 4, 16, 20, 0, 1, 4, 5, 16, 17, 20, 21, 2, 4, 18, 20, 2, 5, 18, 21, 8, 16, 8, 9, 16, 17

Offset: 0

Rémy Sigrist, Jul 06 2024

In other words, we partition n into pairs of fibbinary numbers whose binary expansions have no common 1's and list the corresponding fibbinary numbers to get the n-th row.

			Triangle T(n, k) begins:
  n   n-th row
  --  -----------
   0  0
   1  0, 1
   2  0, 2
   3  1, 2
   4  0, 4
   5  0, 1, 4, 5
   6  2, 4
   7  2, 5
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  1, 2, 9, 10
  12  4, 8
  13  4, 5, 8, 9
  14  4, 10
  15  5, 10
  16  0, 16

Rémy Sigrist, Table of n, a(n) for n = 0..8118 (rows for n = 0..1023 flattened)
Index entries for sequences related to binary expansion of n

See A295989 and A374361 for similar sequences.

Cf. A003714, A037800, A277561, A374355, A374356.

row(n) = { my (r = [0], e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], r = concat([v + y | v <- r], [v + x | v <- r]); break;););); return (r); }

T(n, 0) = 0 iff n is a fibbinary number.
T(n, k) + T(n, A277561(n)-1-k) = n.
T(n, 0) = A374355(n).
T(n, A277561(n)-1) = A374356(n).
Sum_{k = 0..A277561(n)-1} T(n, k) = n * 2^A037800(n).

A374355 a(n) is the least fibbinary number f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 0, 0, 0, 1, 0, 0, 2, 2, 8, 8, 8, 9, 8, 8, 10, 10, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 16, 16, 16, 17, 16, 16, 18, 18, 16, 16, 16, 17, 20, 20, 20, 21, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 4, 4, 4, 5, 0

Offset: 0

Rémy Sigrist, Jul 06 2024

To compute a(n): replace every other bit with zero (starting with the first bit) in each run of consecutive 1's in the binary expansion of n.

			The first terms, in binary and in decimal, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     0      10          0
   3     1      11          1
   4     0     100          0
   5     0     101          0
   6     2     110         10
   7     2     111         10
   8     0    1000          0
   9     0    1001          0
  10     0    1010          0
  11     1    1011          1
  12     4    1100        100
  13     4    1101        100
  14     4    1110        100
  15     5    1111        101
  16     0   10000          0

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A003714, A374354, A374356.

a(n) = { my (v = 0, e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], v += y; break;););); return (v); }

a(n) = A374354(n, 0).
a(n) = n - A374356(n).
a(n) >= 0 with equality iff n is a fibbinary number.

A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 3, 4, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 3, 3, 4, 5, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2

Offset: 0

Gus Wiseman, Jul 02 2025

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 27 are {1,2,4,5}, with maximal sparse subsets {{1,4},{1,5},{2,4},{2,5}}, so a(27) = 4.

For subsets of {1..n} we get A000931 (shifted), maximal case of A000045 (shifted).
This is the maximal case of A245564.
The greatest number whose binary indices are one of these subsets is A374356.
For prime instead of binary indices we have A385215, maximal case of A166469.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A202064 counts subsets containing n with k maximal runs.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000071, A001629, A010049, A044813, A053538, A119900, A202023, A208342, A384177, A384890.

spars[S_]:=Select[Subsets[S],FreeQ[Differences[#],1]&];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
maximize[sys_]:=Complement@@Prepend[Most[Subsets[#]]&/@sys,sys];
Table[Length[maximize[spars[bpe[n]]]],{n,0,100}]

A374396 a(n) is the last term in the n-th row of A374394.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 4, 7, 7, 6, 7, 7, 12, 11, 12, 10, 10, 12, 11, 12, 20, 20, 19, 20, 20, 17, 16, 17, 20, 20, 19, 20, 20, 33, 32, 33, 31, 31, 33, 32, 33, 28, 28, 27, 28, 28, 33, 32, 33, 31, 31, 33, 32, 33, 54, 54, 53, 54, 54, 51, 50, 51, 54, 54, 53, 54, 54, 46

Offset: 0

Rémy Sigrist, Jul 10 2024

a(n) is the greatest number z <= n such that the Zeckendorf representations of z and n-z have no common Fibonacci numbers and when combined together correspond to the lazy Fibonacci representation of n.

Rémy Sigrist, Table of n, a(n) for n = 0..10944
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A022290, A374356, A374394, A374395.

PARI
```
\\ See Links section.
```

a(n) = A022290(A374356(1+A003754(n)), k).

a(n) = n - A374395(n).

A385215 Number of maximal sparse submultisets of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 03 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sparse submultisets of the prime indices of n = 8 are {{},{1},{1,1},{1,1,1}}, with maximization {{1,1,1}}. So a(8) = 1.
The sparse submultisets of the prime indices of n = 462 are {{},{1},{2},{4},{5},{1,4},{2,4},{1,5},{2,5}}, with maximization {{1,4},{1,5},{2,4},{2,5}}, so a(462) = 4.
The prime indices of n together their a(n) maximal sparse submultisets for n = 1, 6, 210, 462, 30030, 46410:
  {}  {1,2}  {1,2,3,4}  {1,2,4,5}  {1,2,3,4,5,6}  {1,2,3,4,6,7}
  ------------------------------------------------------------
  {}   {1}     {1,3}      {1,4}       {2,5}          {1,3,6}
       {2}     {1,4}      {1,5}       {1,3,5}        {1,3,7}
               {2,4}      {2,4}       {1,3,6}        {1,4,6}
                          {2,5}       {1,4,6}        {1,4,7}
                                      {2,4,6}        {2,4,6}
                                                     {2,4,7}

This is the maximal case of A166469.
For binary instead of prime indices we have A384883, maximal case of A245564.
The greatest number whose prime indices are one of these submultisets is A385216.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000045, A000071, A001629, A202064, A316476, A319630, A374356, A384890.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
maxq[els_]:=Select[els,Not[Or@@Table[Divisible[oth,#],{oth,DeleteCases[els,#]}]]&];
Table[Length[maxq[Select[Divisors[n],FreeQ[Differences[prix[#]],1]&]]],{n,30}]

a(n) <= A166469(n).

A377414 a(n) is the largest term of A126684, say b, such that n AND b = b (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 4, 5, 8, 8, 10, 10, 8, 8, 10, 10, 16, 17, 16, 17, 20, 21, 20, 21, 16, 17, 16, 17, 20, 21, 20, 21, 32, 32, 34, 34, 32, 32, 34, 34, 40, 40, 42, 42, 40, 40, 42, 42, 32, 32, 34, 34, 32, 32, 34, 34, 40, 40, 42, 42, 40, 40, 42, 42, 64, 65, 64, 65

Offset: 0

Rémy Sigrist, Oct 27 2024

For any n > 0 with binary expansion (b_1 = 1, b_2, ..., b_k), the binary expansion of a(n) is (c_1, ..., c_k) where c_i = b_i when i is odd, c_i = 0 when i is even.

For any n, the value c = n - a(n) also belongs to A126684 and satisfies n AND c = c (see A377415).

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     2      11         10
   4     4     100        100
   5     5     101        101
   6     4     110        100
   7     5     111        101
   8     8    1000       1000
   9     8    1001       1000
  10    10    1010       1010
  11    10    1011       1010
  12     8    1100       1000
  13     8    1101       1000
  14    10    1110       1010
  15    10    1111       1010

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

See A063694, A063695 and A374356 for similar sequences.

Cf. A000975, A070939, A126684, A371442, A377415.

a(n) = { my (v = 0, x = exponent(n), y); while (n, n -= 2^y = exponent(n); if (x%2 == y%2, v += 2^y;);); return (v); }

a(n) <= n with equality iff n belongs to A126684.
a(a(n)) = a(n).
a(2*n) = 2*a(n).
a(n) = n AND A000975(A070939(n)). - Alan Michael Gómez Calderón, Jun 27 2025

A385216 Greatest Heinz number of a sparse submultiset of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 4, 13, 14, 5, 16, 17, 9, 19, 20, 21, 22, 23, 8, 25, 26, 27, 28, 29, 10, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 21, 43, 44, 9, 46, 47, 16, 49, 50, 51, 52, 53, 27, 55, 56, 57, 58, 59, 20, 61, 62, 63, 64, 65, 33, 67, 68, 69

Offset: 1

Gus Wiseman, Jul 05 2025

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 12 are {1,1,2}, with sparse submultisets {{},{1},{2},{1,1}}, with Heinz numbers {1,2,3,4}, so a(12) = 4.
The prime indices of 36 are {1,1,2,2}, with sparse submultisets {{},{1},{2},{1,1},{2,2}}, with Heinz numbers {1,2,3,4,9}, so a(36) = 9.
The prime indices of 462 are {1,2,4,5}, with sparse submultisets {{},{1},{2},{4},{5},{1,4},{2,4},{1,5},{2,5}}, with Heinz numbers {1,2,3,7,11,14,21,22,33}, so a(462) = 33.

Sparse submultisets are counted by A166469, maximal A385215.
The union is A319630 (Heinz numbers of sparse multisets), complement A104210.
For binary instead of prime indices we have A374356, see A245564, A384883.
A000005 counts divisors (or submultisets of prime indices).
A001222 counts prime factors, distinct A001221.
A051903 gives greatest prime exponent, least A051904, counted by A091602.
A055396 gives least prime index, greatest A061395, counted by A008284.
A056239 adds up prime indices, row sums of A112798.
A212166 ranks partitions with max multiplicity = length, counted by A239964.
A381542 ranks partitions with max part = max multiplicity, counted by A240312.
Cf. A000720, A005117, A122111, A130091, A268193, A316476.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@Select[Divisors[n],FreeQ[Differences[prix[#]],1]&],{n,100}]

a(n) = n iff n belongs to A319630.

cp's OEIS Frontend

A377355 a(n) is the greatest k not yet in the sequence such that A374356(n) = A374356(k).

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A245564 a(n) = Product_{i in row n of A245562} Fibonacci(i+2).

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A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

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A374355 a(n) is the least fibbinary number f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

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A384883 Number of maximal sparse subsets of the binary indices of n, where a set is sparse iff 1 is not a first difference.

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Mathematica

A374396 a(n) is the last term in the n-th row of A374394.

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A385215 Number of maximal sparse submultisets of the prime indices of n, where a multiset is sparse iff 1 is not a first difference.

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A377414 a(n) is the largest term of A126684, say b, such that n AND b = b (where AND denotes the bitwise AND operator).

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