A096111 -id:A096111 - cp's OEIS frontend

A122198 Permutation of natural numbers: a recursed variant of A122155.

0, 1, 2, 3, 4, 7, 6, 5, 8, 15, 14, 13, 12, 9, 10, 11, 16, 31, 30, 29, 28, 25, 26, 27, 24, 17, 18, 19, 20, 23, 22, 21, 32, 63, 62, 61, 60, 57, 58, 59, 56, 49, 50, 51, 52, 55, 54, 53, 48, 33, 34, 35, 36, 39, 38, 37, 40, 47, 46, 45, 44, 41, 42, 43, 64, 127, 126, 125, 124, 121

Offset: 0

Antti Karttunen, Aug 25 2006

nonn

Maps between A096115 and A096111.

Index entries for sequences that are permutations of the natural numbers

Inverse: A122199.

(define (A122198 n) (if (< n 1) n (A122155 (+ (A053644 n) (A122198 (A053645 n))))))

a(0)=0, otherwise a(n) = A122155(A053644(n)+a(A053645(n))).

A122199 Permutation of natural numbers: a recursed variant of A122155.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 13, 14, 15, 12, 11, 10, 9, 16, 25, 26, 27, 28, 31, 30, 29, 24, 21, 22, 23, 20, 19, 18, 17, 32, 49, 50, 51, 52, 55, 54, 53, 56, 61, 62, 63, 60, 59, 58, 57, 48, 41, 42, 43, 44, 47, 46, 45, 40, 37, 38, 39, 36, 35, 34, 33, 64, 97, 98, 99, 100, 103, 102

Offset: 0

Antti Karttunen, Aug 25 2006

nonn

Maps between A096111 and A096115.

Index entries for sequences that are permutations of the natural numbers

Inverse: A122198.

Cf. A053644, A053645, A096111, A096115, A122155.

(define (A122199 n) (if (< n 1) n (let ((m (A122155 n))) (+ (A053644 m) (A122199 (A053645 m))))))

a(0)=0, otherwise a(n) = A053644(A122155(n)) + a(A053645(A122155(n))).

A333492 Position of first appearance of n in A271410 (LCM of binary indices).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 64, 128, 256, 18, 1024, 12, 4096, 66, 20, 32768, 65536, 258, 262144, 24, 68, 1026, 4194304, 132, 16777216, 4098, 67108864, 72, 268435456, 22, 1073741824, 2147483648, 1028, 65538, 80, 264, 68719476736, 262146, 4100, 144, 1099511627776, 70, 4398046511104

Offset: 1

Gus Wiseman, Mar 28 2020

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 sequence together with the corresponding binary expansions and binary indices begins:
      1:                 1 ~ {1}
      2:                10 ~ {2}
      4:               100 ~ {3}
      8:              1000 ~ {4}
     16:             10000 ~ {5}
      6:               110 ~ {2,3}
     64:           1000000 ~ {7}
    128:          10000000 ~ {8}
    256:         100000000 ~ {9}
     18:             10010 ~ {2,5}
   1024:       10000000000 ~ {11}
     12:              1100 ~ {3,4}
   4096:     1000000000000 ~ {13}
     66:           1000010 ~ {2,7}
     20:             10100 ~ {3,5}
  32768:  1000000000000000 ~ {16}
  65536: 10000000000000000 ~ {17}
    258:         100000010 ~ {2,9}

Giovanni Resta, Table of n, a(n) for n = 1..1000

The version for prime indices is A330225.
The version for standard compositions is A333225.
Let q(k) be the binary indices of k:
- The sum of q(k) is A029931(k).
- The elements of q(k) are row k of A048793.
- The product of q(k) is A096111(k).
- The LCM of q(k) is A271410(k).
- The GCD of q(k) is A326674(k).
GCD of prime indices is A289508.
LCM of prime indices is A290103.
LCM of standard compositions is A333226.
Cf. A000120, A066099, A070939, A074761, A076078, A124767, A285572, A324837, A328219, A328451, A331579, A333227.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[LCM@@bpe[n],{n,10000}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

Terms a(23) and beyond from Giovanni Resta, Mar 29 2020

A357186 Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 4, 5, 5, 4, 5, 5, 5, 3, 4, 5, 5, 4, 5, 5, 5, 5, 5, 6, 6, 5, 6, 6, 6, 4, 5, 5, 5, 5, 6, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 3, 4, 5, 5, 5, 6, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 7, 7

Offset: 0

Gus Wiseman, Sep 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 2 + 1 + 1 + 1 + 1 = 6.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
This is the sum of A029837 over the n-th composition in standard order.
Vertex degrees are A133494.
The version for Heinz numbers of partitions is A325033.
Row sums of A357135.
First differences are A357187.
Cf. A000120, A001511, A003963, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[stc/@stc[n]/.List->Plus,{n,0,100}]

a(n) = A029837(A357134(n)).

A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.

Original entry on oeis.org

1, 2, 3, 6, 7, 22, 23, 32, 33, 48, 49, 86, 87, 112, 113, 516, 517, 580, 581, 1110, 1111, 1136, 1137, 1604, 1605, 5206, 5207, 5232, 5233, 5700, 5701, 8212, 8213, 9236, 9237, 13332, 13333, 16386, 16387, 16450, 16451, 17474, 17475, 21570, 21571, 24576, 24577

Offset: 1

Gus Wiseman, Mar 28 2024

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.

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 terms together with their prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    3: {{},{1}}
    6: {{1},{2}}
    7: {{},{1},{2}}
   22: {{1},{2},{3}}
   23: {{},{1},{2},{3}}
   32: {{1,2}}
   33: {{},{1,2}}
   48: {{3},{1,2}}
   49: {{},{3},{1,2}}
   86: {{1},{2},{3},{4}}
   87: {{},{1},{2},{3},{4}}
  112: {{3},{1,2},{4}}
  113: {{},{3},{1,2},{4}}
  516: {{2},{1,3}}
  517: {{},{2},{1,3}}
  580: {{2},{4},{1,3}}
  581: {{},{2},{4},{1,3}}

Without the covering condition we have A371289.
Without squarefree product we have A371292.
Interchanging binary and prime indices gives A371448.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts ordered set partitions, allowing empty sets A000629.
A005117 lists squarefree numbers.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A131689 counts patterns by number of distinct parts.
A302521 lists MM-numbers of set partitions, with empties A302505.
A326701 lists BII-numbers of set partitions.
A368533 lists numbers with squarefree binary indices, prime indices A302478.
Cf. A000040, A001222, A255906, A326782, A371291, A371294, A371447, A371452.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SquareFreeQ[Times @@ bpe[#]]&&normQ[Join@@prix/@bpe[#]]&]

Intersection of A371292 and A371289.

A371443 Numbers whose binary indices are nonprime numbers.

Original entry on oeis.org

1, 8, 9, 32, 33, 40, 41, 128, 129, 136, 137, 160, 161, 168, 169, 256, 257, 264, 265, 288, 289, 296, 297, 384, 385, 392, 393, 416, 417, 424, 425, 512, 513, 520, 521, 544, 545, 552, 553, 640, 641, 648, 649, 672, 673, 680, 681, 768, 769, 776, 777, 800, 801, 808

Offset: 1

Gus Wiseman, Mar 30 2024

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 terms together with their binary expansions and binary indices begin:
    1:          1 ~ {1}
    8:       1000 ~ {4}
    9:       1001 ~ {1,4}
   32:     100000 ~ {6}
   33:     100001 ~ {1,6}
   40:     101000 ~ {4,6}
   41:     101001 ~ {1,4,6}
  128:   10000000 ~ {8}
  129:   10000001 ~ {1,8}
  136:   10001000 ~ {4,8}
  137:   10001001 ~ {1,4,8}
  160:   10100000 ~ {6,8}
  161:   10100001 ~ {1,6,8}
  168:   10101000 ~ {4,6,8}
  169:   10101001 ~ {1,4,6,8}
  256:  100000000 ~ {9}
  257:  100000001 ~ {1,9}
  264:  100001000 ~ {4,9}
  265:  100001001 ~ {1,4,9}
  288:  100100000 ~ {6,9}
  289:  100100001 ~ {1,6,9}
  296:  100101000 ~ {4,6,9}

For powers of 2 instead of nonprime numbers we have A253317.
For prime indices instead of binary indices we have A320628.
For prime instead of nonprime we have A326782.
For composite numbers we have A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A001222, A005117, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],And@@Not/@PrimeQ/@bpe[#]&]

A371444 Numbers whose binary indices are composite numbers.

Original entry on oeis.org

8, 32, 40, 128, 136, 160, 168, 256, 264, 288, 296, 384, 392, 416, 424, 512, 520, 544, 552, 640, 648, 672, 680, 768, 776, 800, 808, 896, 904, 928, 936, 2048, 2056, 2080, 2088, 2176, 2184, 2208, 2216, 2304, 2312, 2336, 2344, 2432, 2440, 2464, 2472, 2560, 2568

Offset: 1

Gus Wiseman, Mar 30 2024

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 terms together with their binary expansions and binary indices begin:
     8:           1000 ~ {4}
    32:         100000 ~ {6}
    40:         101000 ~ {4,6}
   128:       10000000 ~ {8}
   136:       10001000 ~ {4,8}
   160:       10100000 ~ {6,8}
   168:       10101000 ~ {4,6,8}
   256:      100000000 ~ {9}
   264:      100001000 ~ {4,9}
   288:      100100000 ~ {6,9}
   296:      100101000 ~ {4,6,9}
   384:      110000000 ~ {8,9}
   392:      110001000 ~ {4,8,9}
   416:      110100000 ~ {6,8,9}
   424:      110101000 ~ {4,6,8,9}
   512:     1000000000 ~ {10}
   520:     1000001000 ~ {4,10}
   544:     1000100000 ~ {6,10}
   552:     1000101000 ~ {4,6,10}
   640:     1010000000 ~ {8,10}
   648:     1010001000 ~ {4,8,10}
   672:     1010100000 ~ {6,8,10}

For powers of 2 instead of composite numbers we have A253317.
For prime indices we have the even case of A320628.
For prime instead of composite we have A326782.
This is the even case of A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A001222, A005117, A055887, A320629, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],EvenQ[#]&&And@@Not/@PrimeQ/@bpe[#]&]

A371453 Numbers whose binary indices are all squarefree semiprimes.

Original entry on oeis.org

32, 512, 544, 8192, 8224, 8704, 8736, 16384, 16416, 16896, 16928, 24576, 24608, 25088, 25120, 1048576, 1048608, 1049088, 1049120, 1056768, 1056800, 1057280, 1057312, 1064960, 1064992, 1065472, 1065504, 1073152, 1073184, 1073664, 1073696, 2097152, 2097184

Offset: 1

Gus Wiseman, Apr 02 2024

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 terms together with their binary expansions and binary indices begin:
       32:                 100000 ~ {6}
      512:             1000000000 ~ {10}
      544:             1000100000 ~ {6,10}
     8192:         10000000000000 ~ {14}
     8224:         10000000100000 ~ {6,14}
     8704:         10001000000000 ~ {10,14}
     8736:         10001000100000 ~ {6,10,14}
    16384:        100000000000000 ~ {15}
    16416:        100000000100000 ~ {6,15}
    16896:        100001000000000 ~ {10,15}
    16928:        100001000100000 ~ {6,10,15}
    24576:        110000000000000 ~ {14,15}
    24608:        110000000100000 ~ {6,14,15}
    25088:        110001000000000 ~ {10,14,15}
    25120:        110001000100000 ~ {6,10,14,15}
  1048576:  100000000000000000000 ~ {21}

Partitions of this type are counted by A002100, squarefree case of A101048.
For primes instead of squarefree semiprimes we get A326782.
For prime indices instead of binary indices we have A339113, A339112.
Allowing any squarefree numbers gives A368533.
This is the squarefree case of A371454.
A001358 lists squarefree semiprimes, squarefree A006881.
A005117 lists squarefree numbers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A087086, A112798, A296119, A302478, A326031, A367905, A368109, A371450.

M:= 26: # for terms < 2^M
P:= select(isprime, [$2..(M+1)/2]): nP:= nops(P):
S:= select(`<`,{seq(seq(P[i]*P[j],i=1..j-1),j=1..nP)},M+1):
R:= map(proc(s) local i; add(2^(i-1),i=s) end proc, combinat:-powerset(S) minus {{}}):
sort(convert(R,list)); # Robert Israel, Apr 04 2024

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[10000],And@@sqfsemi/@bix[#]&]

def A371453(n): return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')

from math import isqrt
from sympy import primepi, primerange
def A371453(n):
    def f(x,n): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A006881(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024

A371735 Maximal length of a set partition of the binary indices of n into blocks all having the same sum.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Apr 14 2024

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.

If a(n) = k then the binary indices of n (row n of A048793) are k-quanimous (counted by A371783).

			The binary indices of 119 are {1,2,3,5,6,7}, and the set partitions into blocks with the same sum are:
  {{1,7},{2,6},{3,5}}
  {{1,5,6},{2,3,7}}
  {{1,2,3,6},{5,7}}
  {{1,2,3,5,6,7}}
So a(119) = 3.

Set partitions of this type are counted by A035470, A336137.
A version for factorizations is A371733.
Positions of 1's are A371738.
Positions of terms > 1 are A371784.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321452 counts quanimous partitions, ranks A321454.
A326031 gives weight of the set-system with BII-number n.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A038041, A096111, A279787, A305551, A321451, A321455, A322794, A326534, A371731, A371734.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Max[Length/@Select[sps[bix[n]],SameQ@@Total/@#&]],{n,0,100}]

A372431 Positive integers k such that the prime indices of k are disjoint from the binary indices of k.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 10, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 26, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 53, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 84, 86, 89, 92, 93, 94, 96, 97, 98, 101

Offset: 1

Gus Wiseman, May 03 2024

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.

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 binary indices of 65 are {1,7}, and the prime indices are {3,6}, so 65 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     7: {4}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    13: {6}
    16: {1,1,1,1}
The terms together with their binary expansions and binary indices begin:
   1:       1 ~ {1}
   2:      10 ~ {2}
   4:     100 ~ {3}
   7:     111 ~ {1,2,3}
   8:    1000 ~ {4}
   9:    1001 ~ {1,4}
  10:    1010 ~ {2,4}
  11:    1011 ~ {1,2,4}
  12:    1100 ~ {3,4}
  13:    1101 ~ {1,3,4}
  16:   10000 ~ {5}

For subset instead of disjoint we have A372430.
The complement is A372432.
Equal lengths: A071814, zeros of A372441.
Equal sums: A372427, zeros of A372428.
Equal maxima: A372436, zeros of A372442.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A001221, A059893, A096111, A230877, A243055, A304818, A355536, A358136, A372429.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[bix[#],prix[#]]=={}&]

cp's OEIS Frontend

A122198 Permutation of natural numbers: a recursed variant of A122155.

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A122199 Permutation of natural numbers: a recursed variant of A122155.

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A333492 Position of first appearance of n in A271410 (LCM of binary indices).

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Mathematica

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A357186 Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.

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A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.

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A371443 Numbers whose binary indices are nonprime numbers.

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A371444 Numbers whose binary indices are composite numbers.

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A371453 Numbers whose binary indices are all squarefree semiprimes.

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