A053754 -id:A053754 - cp's OEIS frontend

A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

2, 6, 7, 8, 9, 10, 11, 13, 15, 19, 24, 28, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 44, 46, 47, 50, 51, 52, 54, 57, 58, 59, 60, 61, 65, 67, 70, 73, 76, 77, 79, 85, 86, 90, 95, 96, 97, 98, 103, 106, 107, 109, 110, 111, 112, 117, 119, 123, 124, 126, 127, 128, 129

Offset: 1

Gus Wiseman, May 14 2024

nonn

The odd version is A372590.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
          {2}   2  (1)
        {2,3}   6  (2,1)
      {1,2,3}   7  (4)
          {4}   8  (1,1,1)
        {1,4}   9  (2,2)
        {2,4}  10  (3,1)
      {1,2,4}  11  (5)
      {1,3,4}  13  (6)
    {1,2,3,4}  15  (3,2)
      {1,2,5}  19  (8)
        {4,5}  24  (2,1,1,1)
      {3,4,5}  28  (4,1,1)
  {1,2,3,4,5}  31  (11)
          {6}  32  (1,1,1,1,1)
        {1,6}  33  (5,2)
        {2,6}  34  (7,1)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
    {1,2,3,6}  39  (6,2)
        {4,6}  40  (3,1,1,1)
      {1,4,6}  41  (13)
      {2,4,6}  42  (4,2,1)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
Positions of even terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372590.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031215 lists even-indexed primes, odd A031368.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A006141, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[100],EvenQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

A179888 Starting with a(1)=2: if m is a term then also 4m+1 and 4m+2.

Original entry on oeis.org

2, 9, 10, 37, 38, 41, 42, 149, 150, 153, 154, 165, 166, 169, 170, 597, 598, 601, 602, 613, 614, 617, 618, 661, 662, 665, 666, 677, 678, 681, 682, 2389, 2390, 2393, 2394, 2405, 2406, 2409, 2410, 2453, 2454, 2457, 2458, 2469, 2470, 2473, 2474, 2645, 2646, 2649

Offset: 1

Reinhard Zumkeller, Jul 31 2010

nonn

0 -> 01 and 1 -> 10 in binary representation of n;
intersection of A032925 and A053754;
subsequence of A063037;
A000120(a(n))=A023416(a(n))=A070939(n); A070939(a(n))=2*A070939(n).

			__ n | __ bin(n) || ___ bin(a(n)) | base-4(a(n)) | __ a(n)
-----|-----------||---------------|--------------|---------
.. 1 | ....... 1 || .......... 10 | .......... 2 | ..... 2;
.. 2 | ...... 10 || ........ 1001 | ......... 21 | ..... 9;
.. 3 | ...... 11 || ........ 1010 | ......... 22 | .... 10;
.. 4 | ..... 100 || ...... 100101 | ........ 211 | .... 37;
.. 5 | ..... 101 || ...... 100110 | ........ 212 | .... 38;
.. 6 | ..... 110 || ...... 101001 | ........ 221 | .... 41;
.. 7 | ..... 111 || ...... 101010 | ........ 222 | .... 42;
.. 8 | .... 1000 || .... 10010101 | ....... 2111 | ... 149;
.. 9 | .... 1001 || .... 10010110 | ....... 2112 | ... 150;
. 10 | .... 1010 || .... 10011001 | ....... 2121 | ... 153;
. 11 | .... 1011 || .... 10011010 | ....... 2122 | ... 154;
. 12 | .... 1100 || .... 10100101 | ....... 2211 | ... 165;
. 13 | .... 1101 || .... 10100110 | ....... 2212 | ... 166;
. 14 | .... 1110 || .... 10101001 | ....... 2221 | ... 169;
. 15 | .... 1111 || .... 10101010 | ....... 2222 | ... 170;
. 16 | ... 10000 || .. 1001010101 | ...... 21111 | ... 597;
. 17 | ... 10001 || .. 1001010110 | ...... 21112 | ... 598;
. 18 | ... 10010 || .. 1001011001 | ...... 21121 | ... 601;
. 19 | ... 10011 || .. 1001011010 | ...... 21122 | ... 602;
. 20 | ... 10100 || .. 1001100101 | ...... 21211 | ... 613.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Quaternary
Index entries for sequences related to binary expansion of n

Cf. A132679, A196168.

a179888 n = a179888_list !! (n-1)
a179888_list = 2 : f a179888_list where
  f (x:xs) = x' : x'' : f (xs ++ [x',x'']) where x' = 4*x+1; x'' = x' + 1
-- Reinhard Zumkeller, Oct 29 2011

a:= n-> 1+(n mod 2)+`if`(n<2, 0, 4*a(iquo(n, 2))):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 15 2024

Union@ Flatten@ NestList[ {4 # + 1, 4 # + 2} &, 2, 5] (* Robert G. Wilson v, Aug 16 2011 *)

def A179888(n): return ((1<<(n.bit_length()<<1))-1)//3+int(bin(n)[2:],4) # Chai Wah Wu, Jul 16 2024

a(n) = 4*a(floor(n/2)) + n mod 2 + 1 for n>1;

a(n) = SUM((bit(k)+1)*4^k: 0<=k<=L), where bit() and L such that n=SUM(bit(k)*2^k: 0<=k<=L).

A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

Original entry on oeis.org

2, 6, 7, 8, 10, 11, 15, 18, 19, 21, 24, 26, 27, 28, 29, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 65, 70, 71, 72, 74, 76, 78, 79, 81, 84, 86, 87, 89, 91, 95, 96, 98, 101, 104, 105, 106, 107, 108, 111, 112, 113, 114, 116, 117, 122, 126, 128

Offset: 1

Gus Wiseman, May 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.
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 even version is A372589.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {2}   2  (1)
      {2,3}   6  (2,1)
    {1,2,3}   7  (4)
        {4}   8  (1,1,1)
      {2,4}  10  (3,1)
    {1,2,4}  11  (5)
  {1,2,3,4}  15  (3,2)
      {2,5}  18  (2,2,1)
    {1,2,5}  19  (8)
    {1,3,5}  21  (4,2)
      {4,5}  24  (2,1,1,1)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
    {3,4,5}  28  (4,1,1)
  {1,3,4,5}  29  (10)
        {6}  32  (1,1,1,1,1)
      {1,6}  33  (5,2)
      {2,6}  34  (7,1)
      {4,6}  40  (3,1,1,1)
    {1,4,6}  41  (13)
    {3,4,6}  44  (5,1,1)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
Positions of odd terms in A372442, zeros A372436.
The complement is A372589.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
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, A006141, A066208, A160786, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[2,100],OddQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]

Numbers k such that A070939(k) + A061395(k) is odd.

A030301 n-th run has length 2^(n-1).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jean-Paul Delahaye

Cf. A000523, A029837.
Cf. A030300. Partial sums give A079954.
Characteristic function of A053754 (after its initial 0).

[Floor(Log(n)/Log(2)) mod 2: n in [1..100]]; // Vincenzo Librandi, Jun 23 2015

nMax = 7; Table[1 - Mod[n, 2], {n, nMax}, {2^(n-1)}] // Flatten (* Jean-François Alcover, Oct 20 2016 *)
Table[{PadRight[{},2^(n-1),0],PadRight[{},2^n,1]},{n,1,8,2}]//Flatten (* Harvey P. Dale, Apr 12 2023 *)

PARI
```
a(n)=if(n<1,0,1-length(binary(n))%2)
```

a(n)=if(n<1,0,if(n%2==0,-a(n/2)+1,-a((n-1)/2)+1-(((n-1)/2)==0))) /* Ralf Stephan */

def A030301(n): return n.bit_length()&1^1 # Chai Wah Wu, Jan 30 2023

a(n) = A000523(n) mod 2 = (A029837(n+1)+1) mod 2.
a(n) = 0 iff n has an odd number of digits in binary, = 1 otherwise. - Henry Bottomley, Apr 06 2000
a(n) = (1/2)*{1-(-1)^floor(log(n)/log(2))}. - Benoit Cloitre, Nov 22 2001
a(n) = 1-a(floor(n/2)). - Vladeta Jovovic, Aug 04 2003
a(n) = 1 - A030300(n). - Antti Karttunen, Oct 10 2017

A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 12, 15, 16, 17, 20, 21, 29, 32, 36, 42, 43, 45, 46, 47, 48, 51, 53, 54, 55, 59, 60, 61, 63, 64, 65, 66, 67, 68, 71, 73, 78, 79, 80, 81, 84, 89, 91, 93, 94, 95, 97, 99, 101, 105, 110, 111, 113, 114, 115, 116, 118, 119, 121, 122, 125, 127

Offset: 1

Gus Wiseman, May 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.
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 even version is A372587.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {1}   1  ()
            {2}   2  (1)
          {1,2}   3  (2)
            {3}   4  (1,1)
          {1,3}   5  (3)
            {4}   8  (1,1,1)
          {1,4}   9  (2,2)
          {3,4}  12  (2,1,1)
      {1,2,3,4}  15  (3,2)
            {5}  16  (1,1,1,1)
          {1,5}  17  (7)
          {3,5}  20  (3,1,1)
        {1,3,5}  21  (4,2)
      {1,3,4,5}  29  (10)
            {6}  32  (1,1,1,1,1)
          {3,6}  36  (2,2,1,1)
        {2,4,6}  42  (4,2,1)
      {1,2,4,6}  43  (14)
      {1,3,4,6}  45  (3,2,2)
      {2,3,4,6}  46  (9,1)
    {1,2,3,4,6}  47  (15)
          {5,6}  48  (2,1,1,1,1)

Positions of odd terms in A372428, zeros A372427.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372587.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
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, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

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

Numbers k such that A029931(k) + A056239(k) is odd.

A372589 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is even.

Original entry on oeis.org

3, 4, 5, 9, 12, 13, 14, 16, 17, 20, 22, 23, 25, 30, 31, 35, 36, 37, 38, 39, 42, 43, 48, 49, 52, 53, 54, 56, 57, 58, 61, 63, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124

Offset: 1

Gus Wiseman, May 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.
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 odd version is A372588.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1,2}   3  (2)
          {3}   4  (1,1)
        {1,3}   5  (3)
        {1,4}   9  (2,2)
        {3,4}  12  (2,1,1)
      {1,3,4}  13  (6)
      {2,3,4}  14  (4,1)
          {5}  16  (1,1,1,1)
        {1,5}  17  (7)
        {3,5}  20  (3,1,1)
      {2,3,5}  22  (5,1)
    {1,2,3,5}  23  (9)
      {1,4,5}  25  (3,3)
    {2,3,4,5}  30  (3,2,1)
  {1,2,3,4,5}  31  (11)
      {1,2,6}  35  (4,3)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
      {2,3,6}  38  (8,1)
    {1,2,3,6}  39  (6,2)
      {2,4,6}  42  (4,2,1)
    {1,2,4,6}  43  (14)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
Positions of even terms in A372442, zeros A372436.
The complement is A372588.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031215 lists even-indexed primes, odd A031368.
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, A006141, A066207, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[2,100],EvenQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]

Numbers k such that A070939(k) + A061395(k) is even.

A372590 Numbers whose binary weight (A000120) plus bigomega (A001222) is odd.

Original entry on oeis.org

1, 3, 4, 5, 12, 14, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 35, 38, 43, 45, 48, 49, 53, 55, 56, 62, 63, 64, 66, 68, 69, 71, 72, 74, 75, 78, 80, 81, 82, 83, 84, 87, 88, 89, 91, 92, 93, 94, 99, 100, 101, 102, 104, 105, 108, 113, 114, 115, 116, 118, 120

Offset: 1

Gus Wiseman, May 14 2024

nonn

The even version is A372591.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1}   1  ()
      {1,2}   3  (2)
        {3}   4  (1,1)
      {1,3}   5  (3)
      {3,4}  12  (2,1,1)
    {2,3,4}  14  (4,1)
        {5}  16  (1,1,1,1)
      {1,5}  17  (7)
      {2,5}  18  (2,2,1)
      {3,5}  20  (3,1,1)
    {1,3,5}  21  (4,2)
    {2,3,5}  22  (5,1)
  {1,2,3,5}  23  (9)
    {1,4,5}  25  (3,3)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
  {1,3,4,5}  29  (10)
  {2,3,4,5}  30  (3,2,1)
    {1,2,6}  35  (4,3)
    {2,3,6}  38  (8,1)
  {1,2,4,6}  43  (14)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586, complement A372587.
For minimum (A372437) we have A372439, complement A372440.
Positions of odd terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372591.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[100],OddQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

A372587 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is even.

Original entry on oeis.org

6, 7, 10, 11, 13, 14, 18, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 44, 49, 50, 52, 56, 57, 58, 62, 69, 70, 72, 74, 75, 76, 77, 82, 83, 85, 86, 87, 88, 90, 92, 96, 98, 100, 102, 103, 104, 106, 107, 108, 109, 112, 117, 120, 123

Offset: 1

Gus Wiseman, May 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.
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 odd version is A372586.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {2,3}   6  (2,1)
          {1,2,3}   7  (4)
            {2,4}  10  (3,1)
          {1,2,4}  11  (5)
          {1,3,4}  13  (6)
          {2,3,4}  14  (4,1)
            {2,5}  18  (2,2,1)
          {1,2,5}  19  (8)
          {2,3,5}  22  (5,1)
        {1,2,3,5}  23  (9)
            {4,5}  24  (2,1,1,1)
          {1,4,5}  25  (3,3)
          {2,4,5}  26  (6,1)
        {1,2,4,5}  27  (2,2,2)
          {3,4,5}  28  (4,1,1)
        {2,3,4,5}  30  (3,2,1)
      {1,2,3,4,5}  31  (11)
            {1,6}  33  (5,2)
            {2,6}  34  (7,1)
          {1,2,6}  35  (4,3)
          {1,3,6}  37  (12)
          {2,3,6}  38  (8,1)

Positions of even terms in A372428, zeros A372427.
For minimum (A372437) we have A372440, complement A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372586.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
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, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

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

Numbers k such that A029931(k) + A056239(k) is even.

A347704 Number of even-length integer partitions of n with integer alternating product.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 6, 4, 11, 8, 18, 13, 33, 22, 49, 38, 79, 58, 122, 90, 186, 139, 268, 206, 402, 304, 569, 448, 817, 636, 1152, 907, 1612, 1283, 2220, 1791, 3071, 2468, 4162, 3409, 5655, 4634, 7597, 6283, 10171, 8478, 13491, 11336, 17906, 15088, 23513, 20012

Offset: 0

Gus Wiseman, Sep 17 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(2) = 1 through a(9) = 8 partitions:
  (11)  (21)  (22)    (41)    (33)      (61)      (44)        (63)
              (31)    (2111)  (42)      (2221)    (62)        (81)
              (1111)          (51)      (4111)    (71)        (3321)
                              (2211)    (211111)  (2222)      (4221)
                              (3111)              (3221)      (6111)
                              (111111)            (3311)      (222111)
                                                  (4211)      (411111)
                                                  (5111)      (21111111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)

Allowing any alternating product >= 1 gives A000041, reverse A344607.
Allowing any alternating product gives A027187, odd bisection A236914.
The Heinz numbers of these partitions are given by A028260 /\ A347457.
The reverse and reciprocal versions are both A035363.
The multiplicative version (factorizations) is A347438, reverse A347439.
The odd-length instead of even-length version is A347444.
Allowing any length gives A347446.
A034008 counts even-length compositions, ranked by A053754.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A067661, A236913, A304620, A339846, A347437, A347441, A347442, A347445, A347448, A347449, A347454, A347462.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]

A372358 a(n) = n XOR A086893(1+A000523(n)), where XOR is a bitwise-XOR, A003987.

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 2, 5, 4, 7, 6, 1, 0, 3, 2, 5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10, 21, 20, 23, 22, 17, 16, 19, 18, 29, 28, 31, 30, 25, 24, 27, 26, 5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10, 21, 20, 23, 22, 17, 16, 19, 18, 29, 28, 31, 30, 25, 24, 27, 26, 5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15

Offset: 1

Antti Karttunen, May 01 2024

a(n) gives n xored with the unique term of A086893 that has the same binary length as n itself. The binary expansions of the terms of A086893 are of the form 10101...0101 (i.e., alternating 1's and 0's starting and ending with 1) when the binary length is odd, and of the form 110101...0101 (i.e., 1 followed by alternating 1's and 0's, and ending with 1) when the binary length is even. In other words, a(n) is n with its all its even-positioned bits (indexing starts from 0 which stands for the least significant bit) inverted, and additionally also the odd-positioned most significant bit inverted if the number of significant bits is even (i.e., n is a nonzero term of A053754).

			25 in binary is 11001_2, and inverting all the even-positioned bits gives 01100_2, and as A007088(12) = 1100, a(25) = 12.
46 in binary is 101110_2, so we flip all the even-positioned bits (starting from the rightmost, with position 0), and because there are even number of bits in the binary expansion, we flip also the most significant bit, thus we obtain 011011_2, and as A007088(27) = 11011, a(46) = 27.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A000523, A003987, A007088, A053754, A086893 (positions of 0's), A372359, A372360, A372361, A372446.

A000523(n) = logint(n,2);
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3); \\ From A086893
A372358(n) = bitxor(n, A086893(1+A000523(n)));

cp's OEIS Frontend

A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A179888 Starting with a(1)=2: if m is a term then also 4*m+1 and 4*m+2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A030301 n-th run has length 2^(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Formula

A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A372589 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is even.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A372590 Numbers whose binary weight (A000120) plus bigomega (A001222) is odd.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A372587 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is even.

Views

Author

Keywords

A179888 Starting with a(1)=2: if m is a term then also 4m+1 and 4m+2.