A073493 -id:A073493 - cp's OEIS frontend

A356844 Numbers k such that the k-th composition in standard order contains at least one 1. Numbers that are odd or whose binary expansion contains at least two adjacent 1's.

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Sep 02 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.

			The terms, binary expansions, and standard compositions:
   1:       1  (1)
   3:      11  (1,1)
   5:     101  (2,1)
   6:     110  (1,2)
   7:     111  (1,1,1)
   9:    1001  (3,1)
  11:    1011  (2,1,1)
  12:    1100  (1,3)
  13:    1101  (1,2,1)
  14:    1110  (1,1,2)
  15:    1111  (1,1,1,1)
  17:   10001  (4,1)
  19:   10011  (3,1,1)
  21:   10101  (2,2,1)
  22:   10110  (2,1,2)
  23:   10111  (2,1,1,1)
  24:   11000  (1,4)
  25:   11001  (1,3,1)
  26:   11010  (1,2,2)
  27:   11011  (1,2,1,1)
  28:   11100  (1,1,3)
  29:   11101  (1,1,2,1)
  30:   11110  (1,1,1,2)
  31:   11111  (1,1,1,1,1)

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

See link for sequences related to standard compositions.
The case beginning with 1 is A004760, complement A004754.
The complement is A022340.
These compositions are counted by A099036, complement A212804.
The case covering an initial interval is A333217.
The gapless but non-initial version is A356843, unordered A356845.
Cf. A004780, A005408, A055932, A073492, A073493, A132747.

Select[Range[0,100],OddQ[#]||MatchQ[IntegerDigits[#,2],{_,1,1,_}]&]

Union of A005408 and A004780.

A356845 Odd numbers with gapless prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 75, 77, 79, 81, 83, 89, 97, 101, 103, 105, 107, 109, 113, 121, 125, 127, 131, 135, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 191

Offset: 1

Gus Wiseman, Sep 03 2022

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 sequence is gapless if it covers an interval of positive integers.

			The terms together with their prime indices begin:
    1: {}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}

Consists of the odd terms of A073491.
These partitions are counted by A264396.
The strict case is A294674, counted by A136107.
The version for compositions is A356843, counted by A251729.
A001221 counts distinct prime factors, sum A001414.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
Cf. A003963, A034296, A055932, A073493, A107428, A287170, A289508, A325160, A356231, A356234, A356841.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[1,100,2],nogapQ[primeMS[#]]&]

A073487 Squarefree numbers having exactly one prime gap.

Original entry on oeis.org

10, 14, 21, 22, 26, 33, 34, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 106, 111, 114, 115, 118, 119, 122, 123, 129, 133, 134, 138, 141, 142, 145, 146, 154, 155, 158, 159, 161, 165, 166, 174, 177, 178, 183, 185

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

A073484(a(n)) = 1.

			78 is a term, as 78 = 2*3*13 with one gap between 3 and 13.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005117, A073486, A073488, A073489, A073493.

N:= 1000: # to get all terms <= N
Res:= NULL:
for a from 1 to numtheory:-pi(isqrt(N)) do
  for b from a do
    p:= mul(ithprime(i),i=a..b);
    if p > N/ithprime(b+2) then break fi;
    for c from b+2 while p*ithprime(c) <= N do
      for d from c do
        q:= mul(ithprime(i),i=c..d);
        if p*q > N then break fi;
        Res:= Res, p*q;
      od
    od
  od
od:
sort([Res]); # Robert Israel, Apr 20 2017

okQ[n_] := SquareFreeQ[n] && Length[SequencePosition[FactorInteger[n][[All, 1]], {p_?PrimeQ, q_?PrimeQ} /; q != NextPrime[p]]] == 1;
Select[Range[200], okQ] (* Jean-François Alcover, Feb 28 2019 *)

A073494 Numbers having exactly two prime gaps in their factorization.

Original entry on oeis.org

110, 130, 170, 182, 190, 220, 230, 238, 260, 266, 273, 290, 310, 322, 340, 357, 364, 370, 374, 380, 399, 406, 410, 418, 430, 434, 440, 460, 470, 476, 483, 494, 506, 518, 520, 530, 532, 546, 550, 561, 574, 580, 590, 598, 602, 609, 610, 620, 627, 638, 644

Offset: 1

Reinhard Zumkeller, Aug 03 2002

nonn

A073490(a(n)) = 2.

			220 is a term, as 220 = 2*2*5*11 with two gaps: between 2 and 5 and between 5 and 11.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005117, A073492, A073493, A073495, A073488.

a073494 n = a073494_list !! (n-1)
a073494_list = filter ((== 2) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Select[Range[645], Total[pa @@@ Partition[First /@ FactorInteger[#], 2, 1]] == 2 &] (* Jayanta Basu, Jul 01 2013 *)

A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

Original entry on oeis.org

2, 4, 8, 10, 16, 18, 20, 32, 36, 42, 64, 68, 72, 74, 82, 84, 128, 136, 146, 148, 164, 170, 256, 264, 272, 274, 276, 290, 292, 296, 298, 324, 328, 330, 338, 340, 512, 528, 548, 580, 584, 586, 594, 596, 658, 660, 676, 682, 1024, 1040, 1056, 1092, 1096, 1098

Offset: 1

Gus Wiseman, Sep 01 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.

			The terms together with their corresponding standard compositions begin:
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   18: (3,2)
   20: (2,3)
   32: (6)
   36: (3,3)
   42: (2,2,2)
   64: (7)
   68: (4,3)
   72: (3,4)
   74: (3,2,2)
   82: (2,3,2)
   84: (2,2,3)

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

See link for sequences related to standard compositions.
A subset of A022340.
These compositions are counted by A251729.
The unordered version (using Heinz numbers of partitions) is A356845.
A333217 ranks complete compositions.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356841 ranks gapless compositions, counted by A107428.
A356842 ranks non-gapless compositions, counted by A356846.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073491, A073492, A073493, A137921, A356224/A356225.

nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],!MemberQ[stc[#],1]&&nogapQ[stc[#]]&]

Complement of A333217 in A356841.

A356842 Numbers k such that the k-th composition in standard order does not cover an interval of positive integers (not gapless).

Original entry on oeis.org

9, 12, 17, 19, 24, 25, 28, 33, 34, 35, 39, 40, 48, 49, 51, 56, 57, 60, 65, 66, 67, 69, 70, 71, 73, 76, 79, 80, 81, 88, 96, 97, 98, 99, 100, 103, 104, 112, 113, 115, 120, 121, 124, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 144, 145

Offset: 1

Gus Wiseman, Sep 01 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.

			The terms and their corresponding standard compositions begin:
   9: (3,1)
  12: (1,3)
  17: (4,1)
  19: (3,1,1)
  24: (1,4)
  25: (1,3,1)
  28: (1,1,3)
  33: (5,1)
  34: (4,2)
  35: (4,1,1)
  39: (3,1,1,1)
  40: (2,4)
  48: (1,5)
  49: (1,4,1)
  51: (1,3,1,1)
  56: (1,1,4)
  57: (1,1,3,1)
  60: (1,1,1,3)

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

See link for sequences related to standard compositions.
An unordered version is A073492, complement A073491.
These compositions are counted by the complement of A107428.
The complement is A356841.
The gapless but non-initial version is A356843, unordered A356845.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A333217, A356224/A356225.

nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!nogapQ[stc[#]]&]

A356604 Number of integer compositions of n into odd parts covering an initial interval of odd positive integers.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 5, 9, 13, 24, 40, 61, 101, 160, 257, 415, 679, 1103, 1774, 2884, 4656, 7517, 12165, 19653, 31753, 51390, 83134, 134412, 217505, 351814, 569081, 920769, 1489587, 2409992, 3899347, 6309059, 10208628, 16518910, 26729830, 43254212, 69994082

Offset: 0

Gus Wiseman, Aug 30 2022

nonn

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (11)  (111)  (13)    (113)    (1113)    (133)      (1133)
                    (31)    (131)    (1131)    (313)      (1313)
                    (1111)  (311)    (1311)    (331)      (1331)
                            (11111)  (3111)    (11113)    (3113)
                                     (111111)  (11131)    (3131)
                                               (11311)    (3311)
                                               (13111)    (111113)
                                               (31111)    (111131)
                                               (1111111)  (111311)
                                                          (113111)
                                                          (131111)
                                                          (311111)
                                                          (11111111)
The a(9) = 24 compositions:
  (135)  (11133)  (1111113)  (111111111)
  (153)  (11313)  (1111131)
  (315)  (11331)  (1111311)
  (351)  (13113)  (1113111)
  (513)  (13131)  (1131111)
  (531)  (13311)  (1311111)
         (31113)  (3111111)
         (31131)
         (31311)
         (33111)

The case of partitions is A053251, ranked by A356232 and A356603.
These compositions are ranked by the intersection of A060142 and A333217.
This is the odd initial case of A107428.
This is the odd restriction of A107429.
This is the normal/covering case of A324969 (essentially A000045).
The non-initial version is A356605.
A000041 counts partitions, compositions A011782.
A055932 lists numbers with prime indices covering an initial interval.
A066208 lists numbers with all odd prime indices, counted by A000009.
Cf. A001221, A001222, A001227, A005408, A061395, A066205, A073493, A137921, A356224.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[(#+1)/2]&]],{n,0,15}]

More terms from Alois P. Heinz, Sep 01 2022

A137722 Number of numbers not greater than n with exactly one prime gap in their factorization.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 17, 18, 18, 18, 19, 20, 21, 22, 22, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34

Offset: 1

Reinhard Zumkeller, Feb 09 2008

nonn

a(n) > a(n-1) iff A073490(n) = 1;
A137721(n) > a(n) for n < 134;
A137721(n) < a(n) for n > 140.

R. Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A073493.

A162685 Positive integers that are not prime powers and are not divisible by any consecutive primes.

Original entry on oeis.org

10, 14, 20, 21, 22, 26, 28, 33, 34, 38, 39, 40, 44, 46, 50, 51, 52, 55, 56, 57, 58, 62, 63, 65, 68, 69, 74, 76, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 104, 106, 110, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 129, 130, 133, 134, 136, 141, 142

Offset: 1

Leroy Quet, Jul 10 2009

nonn

			220 is factored as 2^2 * 5 * 11. Since both 2 and 5 are not consecutive primes, and 5 and 11 are not consecutive primes (2 and 5 are separated by 3, and 5 and 11 are separated by 7), then 220 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A073492, A073493.

Cf. A000961, A319630.

isA162685 := proc(n) local pfs,i; pfs := numtheory[factorset](n) ; if nops(pfs) <= 1 then RETURN(false) ; else pfs := sort(convert(pfs,list)) ; for i from 2 to nops(pfs) do if op(i,pfs) = nextprime(op(i-1,pfs)) then RETURN(false): fi; od: RETURN(true) ; fi; end: A162685 := proc(n) local a; if n = 1 then 10; else for a from procname(n-1)+1 do if isA162685(a) then RETURN(a) ; fi; od: fi; end: seq(A162685(n),n=1..100) ; # R. J. Mathar, Jul 13 2009

q[n_] := Module[{f = FactorInteger[n]}, Length[f] > 1 && SequenceCount[f[[;; , 1]], {p1_, p2_} /; p2 == NextPrime[p1]] ==  0]; Select[Range[150], q] (* Amiram Eldar, Apr 10 2021 *)

More terms from R. J. Mathar, Jul 13 2009

A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 16, 20, 32, 52, 64, 72, 128, 256, 272, 328, 512, 840, 1024, 1056, 2048, 2320, 4096, 4160, 8192, 10512, 16384, 16512, 17440, 26896, 32768, 65536, 65792, 131072, 135232, 148512, 262144, 262656, 524288, 672800, 1048576, 1049600, 1065088, 1721376

Offset: 1

Gus Wiseman, Sep 24 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

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 intervals begin:
        0: ()
        1: (1)
        2: (2)
        4: (3)
        6: (1,2)
        8: (4)
       16: (5)
       20: (2,3)
       32: (6)
       52: (1,2,3)
       64: (7)
       72: (3,4)
      128: (8)
      256: (9)
      272: (4,5)
      328: (2,3,4)
      512: (10)
      840: (1,2,3,4)

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

See link for sequences related to standard compositions.
These compositions are counted by A001227.
An unordered version is A073485, non-strict A073491 (complement A073492).
The initial version is A164894, non-strict A356843 (unordered A356845).
The non-strict version is A356841, initial A333217, counted by A107428.
A066311 lists gapless numbers.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224, A356842.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Select[Range[0,1000],chQ[stc[#]]&]

cp's OEIS Frontend

A356844 Numbers k such that the k-th composition in standard order contains at least one 1. Numbers that are odd or whose binary expansion contains at least two adjacent 1's.

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A356845 Odd numbers with gapless prime indices.

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A073487 Squarefree numbers having exactly one prime gap.

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A073494 Numbers having exactly two prime gaps in their factorization.

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A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

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A356842 Numbers k such that the k-th composition in standard order does not cover an interval of positive integers (not gapless).

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A356604 Number of integer compositions of n into odd parts covering an initial interval of odd positive integers.

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A137722 Number of numbers not greater than n with exactly one prime gap in their factorization.

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