A328460 -id:A328460 - cp's OEIS frontend

A328608 Numbers whose binary indices have no part circularly followed by a divisor or a multiple.

6, 12, 18, 20, 22, 24, 28, 30, 40, 48, 56, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 104, 108, 110, 112, 114, 116, 118, 120, 124, 126, 132, 144, 148, 156, 160, 172, 176, 180, 188, 192, 196, 204, 208, 212, 220, 224, 236, 240, 244, 252, 258, 264

Offset: 1

Gus Wiseman, Oct 25 2019

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.
Circularity means the last part is followed by the first.
Note that this is a somewhat degenerate case, as a part could only be followed by a divisor if it is the last part followed by the first.

			The sequence of terms together with their binary expansions and binary indices begins:
    6:       110 ~ {2,3}
   12:      1100 ~ {3,4}
   18:     10010 ~ {2,5}
   20:     10100 ~ {3,5}
   22:     10110 ~ {2,3,5}
   24:     11000 ~ {4,5}
   28:     11100 ~ {3,4,5}
   30:     11110 ~ {2,3,4,5}
   40:    101000 ~ {4,6}
   48:    110000 ~ {5,6}
   56:    111000 ~ {4,5,6}
   66:   1000010 ~ {2,7}
   68:   1000100 ~ {3,7}
   70:   1000110 ~ {2,3,7}
   72:   1001000 ~ {4,7}
   76:   1001100 ~ {3,4,7}
   78:   1001110 ~ {2,3,4,7}
   80:   1010000 ~ {5,7}
   82:   1010010 ~ {2,5,7}
   84:   1010100 ~ {3,5,7}

The composition version is A328599.
The necklace composition version is A328601.
Compositions with no consecutive divisors or multiples are A328508.
Numbers whose binary indices are pairwise indivisible are A326704.
Cf. A000031, A318726, A318729, A328171, A328460, A328593, A328598, A328600, A328603.

Select[Range[100],!MatchQ[Append[Join@@Position[Reverse[IntegerDigits[#,2]],1],1+IntegerExponent[#,2]],{_,x_,y_,_}/;Divisible[x,y]||Divisible[y,x]]&]

A328674 Numbers whose distinct prime indices have no consecutive divisible parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 64, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 113, 119, 121, 123, 125, 127, 128, 131, 135

Offset: 1

Gus Wiseman, Oct 29 2019

nonn

First differs from A316476 in having 105, with prime indices {2, 3, 4}.

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 sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
For example, 45 is in the sequence because its distinct prime indices are {2,3} and 2 is not a divisor of 3.

These are the Heinz numbers of the partitions counted by A328675.
The strict version is A328603.
Partitions without consecutive divisibilities are A328171.
Compositions without consecutive divisibilities are A328460.
Cf. A056239, A112798, A316476, A318729, A328335, A328336, A328593, A328598.

Select[Range[100],!MatchQ[PrimePi/@First/@FactorInteger[#],{_,x_,y_,_}/;Divisible[y,x]]&]

A328675 Number of integer partitions of n with no two distinct consecutive parts divisible.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 13, 13, 22, 23, 30, 36, 50, 54, 77, 85, 113, 135, 170, 194, 256, 303, 369, 440, 545, 640, 792, 931, 1132, 1347, 1616, 1909, 2295, 2712, 3225, 3799, 4519, 5310, 6278, 7365, 8675, 10170, 11928, 13940, 16314, 19046, 22223, 25856

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

			The a(1) = 1 through a(10) = 9 partitions (A = 10).
  1  2   3    4     5      6       7        8         9          A
     11  111  22    32     33      43       44        54         55
              1111  11111  222     52       53        72         64
                           111111  322      332       333        73
                                   1111111  2222      432        433
                                            11111111  522        532
                                                      3222       3322
                                                      111111111  22222
                                                                 1111111111

The Heinz numbers of these partitions are given by A328674.
The case involving all consecutive parts (not just distinct) is A328171.
The version for relative primality instead of divisibility is A328187.
Partitions with all consecutive parts divisible are A003238.
Compositions without consecutive divisibilities are A328460.
Cf. A305148, A316476, A318726, A328172, A328508, A328593, A328598, A328603.

Table[Length[Select[IntegerPartitions[n],!MatchQ[Union[#],{_,x_,y_,_}/;Divisible[y,x]]&]],{n,0,30}]

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A328608 Numbers whose binary indices have no part circularly followed by a divisor or a multiple.

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A328674 Numbers whose distinct prime indices have no consecutive divisible parts.

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A328675 Number of integer partitions of n with no two distinct consecutive parts divisible.

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Mathematica