A359402 -id:A359402 - cp's OEIS frontend

A372438 Least binary index equals greatest prime index.

6, 18, 20, 54, 56, 60, 100, 162, 168, 176, 180, 280, 300, 392, 416, 486, 500, 504, 528, 540, 840, 880, 900, 1088, 1176, 1232, 1248, 1400, 1458, 1500, 1512, 1584, 1620, 1936, 1960, 2080, 2432, 2500, 2520, 2640, 2700, 2744, 2912, 3264, 3528, 3696, 3744, 4200

Offset: 1

Gus Wiseman, May 04 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.
Are there any squarefree terms > 6?

			The binary indices of 60 are {3,4,5,6}, the prime indices are {1,1,2,3}, and 3 = 3, so 60 is in the sequence.
The terms together with their prime indices begin:
     6: {1,2}
    18: {1,2,2}
    20: {1,1,3}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
   100: {1,1,3,3}
   162: {1,2,2,2,2}
   168: {1,1,1,2,4}
   176: {1,1,1,1,5}
   180: {1,1,2,2,3}
   280: {1,1,1,3,4}
   300: {1,1,2,3,3}
The terms together with their binary expansions and binary indices begin:
     6:            110 ~ {2,3}
    18:          10010 ~ {2,5}
    20:          10100 ~ {3,5}
    54:         110110 ~ {2,3,5,6}
    56:         111000 ~ {4,5,6}
    60:         111100 ~ {3,4,5,6}
   100:        1100100 ~ {3,6,7}
   162:       10100010 ~ {2,6,8}
   168:       10101000 ~ {4,6,8}
   176:       10110000 ~ {5,6,8}
   180:       10110100 ~ {3,5,6,8}
   280:      100011000 ~ {4,5,9}
   300:      100101100 ~ {3,4,6,9}

Same length: A071814, zeros of A372441.
Same sum: A372427, zeros of A372428.
Same 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, A014499, A030101, A061712, A318283, A355536, A359401, A359402, A372429-A372433.

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[1000],Min[bix[#]]==Max[prix[#]]&]

A001511(a(n)) = A061395(a(n)).

A372440 Numbers k such that the least binary index of k plus the least prime index of k is even.

Original entry on oeis.org

4, 5, 11, 12, 16, 17, 20, 23, 25, 28, 31, 35, 36, 41, 44, 47, 48, 52, 55, 59, 60, 64, 65, 67, 68, 73, 76, 80, 83, 84, 85, 92, 95, 97, 100, 103, 108, 109, 112, 115, 116, 121, 124, 125, 127, 132, 137, 140, 143, 144, 145, 148, 149, 155, 156, 157, 164, 167, 172

Offset: 1

Gus Wiseman, May 06 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 terms (center), their binary indices (left), and their prime indices (right) begin:
          {3}   4  (1,1)
        {1,3}   5  (3)
      {1,2,4}  11  (5)
        {3,4}  12  (2,1,1)
          {5}  16  (1,1,1,1)
        {1,5}  17  (7)
        {3,5}  20  (3,1,1)
    {1,2,3,5}  23  (9)
      {1,4,5}  25  (3,3)
      {3,4,5}  28  (4,1,1)
  {1,2,3,4,5}  31  (11)
      {1,2,6}  35  (4,3)
        {3,6}  36  (2,2,1,1)
      {1,4,6}  41  (13)
      {3,4,6}  44  (5,1,1)
  {1,2,3,4,6}  47  (15)
        {5,6}  48  (2,1,1,1,1)
      {3,5,6}  52  (6,1,1)
  {1,2,3,5,6}  55  (5,3)
  {1,2,4,5,6}  59  (17)
    {3,4,5,6}  60  (3,2,1,1)
          {7}  64  (1,1,1,1,1,1)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
Positions of even terms in A372437.
The complement is 1 followed by A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
A003963 gives product of prime indices, binary A096111.
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, A061712, A174090, A243055, A359402, A359495, A372429, A372430, A372431, A372432, A372438, A372471.

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],EvenQ[Min[bix[#]]+Min[prix[#]]]&]

A372515 Irregular triangle read by rows where row n lists the positions of zeros in the reversed binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 26 2024

			The reversed binary expansion of 100 is (0,0,1,0,0,1,1), with zeros at positions {1,2,4,5}, so row 100 is (1,2,4,5).
Triangle begins:
   1:
   2: 1
   3:
   4: 1 2
   5: 2
   6: 1
   7:
   8: 1 2 3
   9: 2 3
  10: 1 3
  11: 3
  12: 1 2
  13: 2
  14: 1
  15:
  16: 1 2 3 4

Row lengths are A023416, partial sums A059015.
For ones instead of zeros we have A048793, lengths A000120, sums A029931.
Row sums are A359400, non-reversed A359359.
Same as A368494 but with empty rows () instead of (0).
A003714 lists numbers with no successive binary indices.
A030190 gives binary expansion, reverse A030308.
A039004 lists the positions of zeros in A345927.
Cf. A069010, A070939, A073642, A230877, A344618, A359402, A359495.

Table[Join@@Position[Reverse[IntegerDigits[n,2]],0],{n,30}]

A359496 Nonnegative integers whose sum of positions of 1's in their binary expansion is less than the sum of positions of 1's in their reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20, 22, 24, 25, 26, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 64, 66, 68, 72, 74, 76, 80, 81, 82, 84, 86, 88, 89, 90, 92, 94, 96, 97, 98, 100, 101, 102, 104, 105, 106

Offset: 1

Gus Wiseman, Jan 18 2023

First differs from A161602 in lacking 70, with binary expansion (1,0,0,0,1,1,0), positions of 1's 1 + 5 + 6 = 12, reversed 2 + 3 + 7 = 12.

			The initial terms, binary expansions, and positions of 1's are:
    2:      10 ~ {2}
    4:     100 ~ {3}
    6:     110 ~ {2,3}
    8:    1000 ~ {4}
   10:    1010 ~ {2,4}
   12:    1100 ~ {3,4}
   13:    1101 ~ {1,3,4}
   14:    1110 ~ {2,3,4}
   16:   10000 ~ {5}
   18:   10010 ~ {2,5}
   20:   10100 ~ {3,5}
   22:   10110 ~ {2,3,5}
   24:   11000 ~ {4,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}
   28:   11100 ~ {3,4,5}
   29:   11101 ~ {1,3,4,5}
   30:   11110 ~ {2,3,4,5}

The opposite version is A359401.
Indices of negative terms in A359495; indices of 0's are A359402.
A030190 gives binary expansion, reverse A030308.
A070939 counts binary digits.
A230877 adds up positions of 1's in binary expansion, reverse A029931.
A326669 lists numbers with integer mean position of a 1 in binary expansion.
A358194 counts partitions by sum of partial sums, compositions A053632.
Cf. A000120, A048793, A051293, A222955, A231204, A291166, A304818, A326672, A326673, A359043.

Select[Range[100],Total[Accumulate[IntegerDigits[#,2]]]>Total[Accumulate[Reverse[IntegerDigits[#,2]]]]&]

A230877(a(n)) < A029931(a(n)).

cp's OEIS Frontend

A372438 Least binary index equals greatest prime index.

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A372440 Numbers k such that the least binary index of k plus the least prime index of k is even.

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A372515 Irregular triangle read by rows where row n lists the positions of zeros in the reversed binary expansion of n.

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A359496 Nonnegative integers whose sum of positions of 1's in their binary expansion is less than the sum of positions of 1's in their reversed binary expansion, where positions in a sequence are read starting with 1 from the left.

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