A291166 -id:A291166 - cp's OEIS frontend

A338552 Non-powers of primes whose prime indices have a common divisor > 1.

21, 39, 57, 63, 65, 87, 91, 111, 115, 117, 129, 133, 147, 159, 171, 183, 185, 189, 203, 213, 235, 237, 247, 259, 261, 267, 273, 299, 301, 303, 305, 319, 321, 325, 333, 339, 351, 365, 371, 377, 387, 393, 399, 417, 427, 441, 445, 453, 477, 481, 489, 497, 507

Offset: 1

Gus Wiseman, Nov 03 2020

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.

Also Heinz numbers of non-constant, non-relatively prime partitions. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     21: {2,4}      183: {2,18}       305: {3,18}
     39: {2,6}      185: {3,12}       319: {5,10}
     57: {2,8}      189: {2,2,2,4}    321: {2,28}
     63: {2,2,4}    203: {4,10}       325: {3,3,6}
     65: {3,6}      213: {2,20}       333: {2,2,12}
     87: {2,10}     235: {3,15}       339: {2,30}
     91: {4,6}      237: {2,22}       351: {2,2,2,6}
    111: {2,12}     247: {6,8}        365: {3,21}
    115: {3,9}      259: {4,12}       371: {4,16}
    117: {2,2,6}    261: {2,2,10}     377: {6,10}
    129: {2,14}     267: {2,24}       387: {2,2,14}
    133: {4,8}      273: {2,4,6}      393: {2,32}
    147: {2,4,4}    299: {6,9}        399: {2,4,8}
    159: {2,16}     301: {4,14}       417: {2,34}
    171: {2,2,8}    303: {2,26}       427: {4,18}

A318978 allows prime powers, counted by A018783, with complement A289509.
A327685 allows nonprime prime powers.
A338330 is the coprime instead of relatively prime version.
A338554 counts the partitions with these Heinz numbers.
A338555 is the complement.
A000740 counts relatively prime compositions.
A000961 lists powers of primes, with complement A024619.
A051424 counts pairwise coprime or singleton partitions.
A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.
A291166 ranks relatively prime compositions, with complement A291165.
A302696 gives the Heinz numbers of pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.

Select[Range[100],!(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]

Equals A024619 /\ A318978.

Complement of A000961 \/ A289509.

A338555 Numbers that are either a power of a prime or have relatively prime prime indices.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Gus Wiseman, Nov 03 2020

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.

Also Heinz numbers of partitions either constant or relatively prime (A338553). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

A327534 uses primes instead of prime powers.
A338331 is the pairwise coprime version, with complement A338330.
A338552 is the complement.
A338553 counts the partitions with these Heinz numbers.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A000961 lists powers of primes.
A018783 counts partitions whose prime indices are not relatively prime, with Heinz numbers A318978.
A051424 counts pairwise coprime or singleton partitions.
A291166 ranks relatively prime compositions, with complement A291165.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000740, A056239, A108572, A112798, A302569, A302796, A327685, A328677.

Select[Range[100],#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1&]

Equals A000961 \/ A289509.

Complement of A024619 /\ A318978.

A327368 The positions of ones in the reversed binary expansion of n have integer mean and integer geometric mean.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 130, 256, 257, 512, 1024, 2048, 2084, 2316, 4096, 8192, 16384, 32768, 32776, 32777, 65536, 131072, 131074, 131200, 131457, 131462, 133390, 165920, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 8388640, 8388897, 8390688, 8519840

Offset: 1

Gus Wiseman, Sep 27 2019

nonn

			The sequence of terms together with their binary indices begins:
  2      {2}
  4      {3}
  8      {4}
  16     {5}
  32     {6}
  64     {7}
  128    {8}
  130    {2,8}
  256    {9}
  257    {1,9}
  512    {10}
  1024   {11}
  2048   {12}
  2084   {3,6,12}
  2316   {3,4,9,12}
  4096   {13}
  8192   {14}
  16384  {15}
  32768  {16}
  32776  {4,16}

A superset of A327777.
Numbers whose binary indices have integer mean: A326669
Numbers whose binary indices have integer geometric mean: A326673
Cf. A000120, A029931, A048793, A070939, A096111, A291166, A295235, A326029, A326643, A326699/A326700, A327777.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

ok(n)={my(s=0,p=1,k=0); for(i=0, logint(n,2), if(bittest(n,i), s+=i+1; p*=i+1; k++)); s%k==0 && ispower(p,k)}
{ for(n=1, 10^7, if(ok(n), print1(n, ", "))) } \\ Andrew Howroyd, Sep 29 2019

a(33)-a(40) from Andrew Howroyd, Sep 29 2019

A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

Original entry on oeis.org

2, 257, 8519971, 36574494881, 140739702949921, 140773995710729, 140774004099109

Offset: 1

Gus Wiseman, Sep 27 2019

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.

Conjecture: This sequence is infinite.

			The initial terms together with their binary indices:
                2: {2}
              257: {1,9}
          8519971: {1,2,6,9,18,24}
      36574494881: {1,6,8,16,18,27,32,36}
  140739702949921: {1,6,12,27,32,48}
  140773995710729: {1,4,9,12,18,32,36,48}
  140774004099109: {1,3,6,12,18,24,32,36,48}

Wikipedia, Geometric mean

A subset of A327368.
The binary weight of prime(n) is A014499(n), with binary length A035100(n).
Heinz numbers of partitions with integer mean: A316413.
Heinz numbers of partitions with integer geometric mean: A326623.
Heinz numbers with both: A326645.
Subsets with integer mean: A051293
Subsets with integer geometric mean: A326027
Subsets with both: A326643
Partitions with integer mean: A067538
Partitions with integer geometric mean: A067539
Partitions with both: A326641
Strict partitions with integer mean: A102627
Strict partitions with integer geometric mean: A326625
Strict partitions with both: A326029
Factorizations with integer mean: A326622
Factorizations with integer geometric mean: A326028
Factorizations with both: A326647
Numbers whose binary indices have integer mean: A326669
Numbers whose binary indices have integer geometric mean: A326673
Numbers whose binary indices have both: A327368
Cf. A000120, A029931, A034797, A048793, A070939, A096111, A291166, A326031, A326644, A326699/A326700.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Prime[Range[1000]],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

a(4)-a(7) from Giovanni Resta, Dec 01 2019

A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 13, 14, 15, 16, 31, 32, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 127, 128, 136, 138, 139, 141, 142, 143, 162, 163, 168, 170, 171, 173, 174, 175, 177, 181, 182, 183, 184

Offset: 1

Gus Wiseman, Nov 15 2021

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.

			The terms and corresponding compositions begin:
      0: ()              36: (3,3)           54: (1,2,1,2)
      1: (1)             37: (3,2,1)         55: (1,2,1,1,1)
      2: (2)             38: (3,1,2)         57: (1,1,3,1)
      3: (1,1)           39: (3,1,1,1)       58: (1,1,2,2)
      4: (3)             41: (2,3,1)         59: (1,1,2,1,1)
      7: (1,1,1)         42: (2,2,2)         60: (1,1,1,3)
      8: (4)             43: (2,2,1,1)       61: (1,1,1,2,1)
     10: (2,2)           44: (2,1,3)         62: (1,1,1,1,2)
     11: (2,1,1)         45: (2,1,2,1)       63: (1,1,1,1,1,1)
     13: (1,2,1)         46: (2,1,1,2)       64: (7)
     14: (1,1,2)         47: (2,1,1,1,1)    127: (1,1,1,1,1,1,1)
     15: (1,1,1,1)       50: (1,3,2)        128: (8)
     16: (5)             51: (1,3,1,1)      136: (4,4)
     31: (1,1,1,1,1)     52: (1,2,3)        138: (4,2,2)
     32: (6)             53: (1,2,2,1)      139: (4,2,1,1)

Looking at length instead of parts gives A096199.
These composition are counted by A100346.
A version counting subsets instead of compositions is A125297.
An unordered version is A326841, counted by A018818.
A011782 counts compositions.
A316413 ranks partitions with sum divisible by length, counted by A067538.
A319333 ranks partitions with sum equal to lcm, counted by A074761.
Cf. A003242, A056239, A238279, A326836, A326842.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Permutations are ranked by A333218.
- Relatively prime compositions are ranked by A291166*, complement A291165.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],#==0||Divisible[Total[stc[#]],LCM@@stc[#]]&]

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

A338552 Non-powers of primes whose prime indices have a common divisor > 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A338555 Numbers that are either a power of a prime or have relatively prime prime indices.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A327368 The positions of ones in the reversed binary expansion of n have integer mean and integer geometric mean.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula