cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A329136 Number of integer partitions of n whose augmented differences are an aperiodic word.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 10, 14, 19, 28, 40, 53, 75, 99, 131, 172, 226, 294, 380, 488, 617, 787, 996, 1250, 1565, 1953, 2425, 3003, 3705, 4559, 5589, 6836, 8329, 10132, 12292, 14871, 17950, 21629, 25988, 31169, 37306, 44569, 53139, 63247, 75133, 89111, 105515, 124737
Offset: 0

Views

Author

Gus Wiseman, Nov 09 2019

Keywords

Comments

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
A sequence is aperiodic if its cyclic rotations are all different.

Examples

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)  (3)    (4)      (5)        (6)          (7)
            (2,1)  (2,2)    (4,1)      (3,3)        (4,3)
                   (3,1)    (2,2,1)    (4,2)        (5,2)
                   (2,1,1)  (3,1,1)    (5,1)        (6,1)
                            (2,1,1,1)  (2,2,2)      (3,2,2)
                                       (3,2,1)      (3,3,1)
                                       (4,1,1)      (4,2,1)
                                       (2,2,1,1)    (5,1,1)
                                       (3,1,1,1)    (2,2,2,1)
                                       (2,1,1,1,1)  (3,2,1,1)
                                                    (4,1,1,1)
                                                    (2,2,1,1,1)
                                                    (3,1,1,1,1)
                                                    (2,1,1,1,1,1)
With augmented differences:
  (1)  (2)  (3)    (4)      (5)        (6)          (7)
            (2,1)  (1,2)    (4,1)      (1,3)        (2,3)
                   (3,1)    (1,2,1)    (3,2)        (4,2)
                   (2,1,1)  (3,1,1)    (5,1)        (6,1)
                            (2,1,1,1)  (1,1,2)      (1,3,1)
                                       (2,2,1)      (2,1,2)
                                       (4,1,1)      (3,2,1)
                                       (1,2,1,1)    (5,1,1)
                                       (3,1,1,1)    (1,1,2,1)
                                       (2,1,1,1,1)  (2,2,1,1)
                                                    (4,1,1,1)
                                                    (1,2,1,1,1)
                                                    (3,1,1,1,1)
                                                    (2,1,1,1,1,1)

Crossrefs

The Heinz numbers of these partitions are given by A329133.
The periodic version is A329143.
The non-augmented version is A329137.
Aperiodic binary words are A027375.
Aperiodic compositions are A000740.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose differences of prime indices are aperiodic are A329135.
Numbers whose prime signature is aperiodic are A329139.
Cf. A152061, A325351, A325356, A329132, A329134, A329139, A329140.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
aug[y_]:=Table[If[i

Formula

a(n) + A329143(n) = A000041(n).

A383507 Number of Wilf and conjugate Wilf integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 20, 27, 30, 31, 40, 50, 56, 68, 76, 86, 112, 126, 139, 170, 197, 216, 251, 297, 317, 378, 411, 466, 521, 607, 621, 745, 791, 892, 975, 1123, 1163, 1366, 1439, 1635, 1757, 2021, 2080, 2464, 2599, 2882, 3116, 3572, 3713
Offset: 0

Views

Author

Gus Wiseman, May 14 2025

Keywords

Comments

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

Examples

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Links

Crossrefs

A048768 gives Look-and-Say fixed points, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences.
Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#] && UnsameQ@@DeleteCases[Differences[Append[#,0]],0]&]],{n,0,30}]

Formula

These partitions have Heinz numbers A130091 /\ A383512.

A329133 Numbers whose augmented differences of prime indices are an aperiodic sequence.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74
Offset: 1

Views

Author

Gus Wiseman, Nov 09 2019

Keywords

Comments

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
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 finite sequence is aperiodic if its cyclic rotations are all different.

Examples

			The sequence of terms together with their augmented differences of prime indices begins:
    1: ()
    2: (1)
    3: (2)
    5: (3)
    6: (2,1)
    7: (4)
    9: (1,2)
   10: (3,1)
   11: (5)
   12: (2,1,1)
   13: (6)
   14: (4,1)
   17: (7)
   18: (1,2,1)
   19: (8)
   20: (3,1,1)
   21: (3,2)
   22: (5,1)
   23: (9)
   24: (2,1,1,1)

Crossrefs

Complement of A329132.
These are the Heinz numbers of the partitions counted by A329136.
Aperiodic binary words are A027375.
Aperiodic compositions are A000740.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is aperiodic are A329139.
Numbers whose differences of prime indices are aperiodic are A329135.
Cf. A056239, A112798, A121016, A124010, A152061, A246029, A325351, A325389, A329134, A329140.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
aug[y_]:=Table[If[i

A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76
Offset: 1

Views

Author

Gus Wiseman, Mar 23 2021

Keywords

Comments

Also called log-concave-up partitions.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The prime indices of 60 are {1,1,2,3}, with first quotients (1,2,3/2), so 60 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}

Links

Crossrefs

The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A304678.
For differences instead of quotients we have A325360 (count: A240026).
These partitions are counted by A342523 (strict: A342516, ordered: A342492).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342086 counts strict chains of divisors with strictly increasing quotients.
Cf. A000005, A002843, A056239, A067824, A112798, A124010, A130091, A238710, A253249, A325351, A325352, A342191.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],LessEqual@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 2, 1, 7, 2, 8, 3, 3, 5, 9, 2, 3, 6, 2, 4, 10, 2, 11, 1, 4, 7, 3, 2, 12, 8, 5, 3, 13, 3, 14, 5, 2, 9, 15, 2, 4, 3, 6, 6, 16, 2, 3, 4, 7, 10, 17, 2, 18, 11, 3, 1, 4, 4, 19, 7, 8, 3, 20, 2, 21, 12, 2, 8, 4, 5, 22, 3, 2
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2022

Keywords

Comments

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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

Examples

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 3.

Links

Crossrefs

Crossrefs found in the link are not repeated here.
Prepending 1 to the positions of 1's gives A000079.
Positions of first appearances are A008578.
Positions of 2's are A065119.
The non-augmented version is A286470, also A355526.
The non-augmented minimal version is A355524, also A355525.
The minimal version is A355531.
Row maxima of A355534, which has Heinz number A325351.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355533, A355536.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A356958 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (b-a+1, ..., y-a+1, z-a+1).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 1, 2, 4, 2, 1, 1, 1, 2, 2, 1, 3, 3, 5, 1, 1, 2, 1, 6, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 4, 7, 2, 1, 2, 2, 8, 5, 1, 1, 3, 2, 4, 1, 5, 1, 2, 9, 1, 1, 1, 2, 1, 3, 3, 6, 1, 6, 2, 2, 2, 3, 1, 1, 4, 7, 10, 1, 2, 3, 11, 1, 3, 1, 1, 1, 1, 1, 4, 2, 5
Offset: 1

Views

Author

Gus Wiseman, Dec 27 2022

Keywords

Comments

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.

Examples

			Triangle begins:
   1:   .
   2:   .
   3:   .
   4:   1
   5:   .
   6:   2
   7:   .
   8:  1 1
   9:   1
  10:   3
  11:   .
  12:  1 2
  13:   .
  14:   4
  15:   2
  16: 1 1 1
For example, the prime indices of 315 are (2,2,3,4), so row 315 is (2,3,4) - 2 + 1 = (1,2,3).

Crossrefs

Row lengths are A001222(n) - 1.
Indices of empty rows are A008578.
Even bisection is A112798.
Heinz numbers of rows are A246277.
An opposite version is A358172, Heinz numbers A358195.
Row sums are A359358(n) + A001222(n) - 1.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
Cf. A055396, A124010, A241916, A253565, A325351, A325352, A326844, A355534, A355536, A358137.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,{},1-First[primeMS[n]]+Rest[primeMS[n]]],{n,100}]

A358172 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (z-a+1, z-b+1, ..., z-y+1).

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 2, 4, 2, 1, 1, 1, 2, 1, 3, 3, 3, 5, 2, 2, 2, 1, 6, 1, 1, 4, 4, 3, 2, 1, 1, 1, 1, 4, 7, 2, 2, 2, 1, 8, 5, 3, 3, 3, 4, 3, 5, 5, 2, 2, 9, 2, 2, 2, 2, 1, 3, 1, 6, 6, 6, 2, 1, 1, 3, 4, 4, 4, 7, 10, 3, 3, 2, 11, 3, 3, 1, 1, 1, 1, 1, 4, 5, 4
Offset: 1

Views

Author

Gus Wiseman, Dec 20 2022

Keywords

Comments

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.

Examples

			Triangle begins:
   1:   .
   2:   .
   3:   .
   4:   1
   5:   .
   6:   2
   7:   .
   8:  1 1
   9:   1
  10:   3
  11:   .
  12:  2 2
  13:   .
  14:   4
  15:   2
  16: 1 1 1
  17:   .
  18:  2 1
  19:   .
  20:  3 3
For example, the prime indices of 900 are (1,1,2,2,3,3), so row 900 is 3 - (1,1,2,2,3) + 1 = (3,3,2,2,1).

Crossrefs

Row lengths are A001222(n) - 1.
Indices of empty rows are A008578.
Even-indexed rows have sums A243503.
Row sums are A326844(n) + A001222(n) - 1.
An opposite version is A356958, Heinz numbers A246277.
Heinz numbers of the rows are A358195, even bisection A241916.
A112798 list prime indices, sum A056239.
A243055 subtracts the least prime index from the greatest.
Cf. A055396, A124010, A253565, A325351, A325352, A355534, A355536, A358137.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,{},1+Last[primeMS[n]]-Most[primeMS[n]]],{n,100}]

A325591 Number of compositions of n with circular differences all equal to 1, 0, or -1.

Original entry on oeis.org

1, 2, 4, 6, 11, 15, 27, 43, 68, 116, 189, 311, 519, 860, 1433, 2380, 3968, 6613, 11018, 18374, 30633, 51089, 85208, 142113, 237055, 395409, 659576, 1100262, 1835382, 3061711, 5107445, 8520122, 14213135, 23710173, 39553138, 65982316, 110071459, 183620990, 306316328
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

Examples

			The a(1) = 1 through a(6) = 15 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (22)    (23)     (33)
             (21)   (112)   (32)     (222)
             (111)  (121)   (122)    (1122)
                    (211)   (212)    (1212)
                    (1111)  (221)    (1221)
                            (1112)   (2112)
                            (1121)   (2121)
                            (1211)   (2211)
                            (2111)   (11112)
                            (11111)  (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)

Links

Crossrefs

Row sums of A309931.
Cf. A000079, A008965, A034297, A173258, A325351, A325551, A325553, A325558, A325589, A325590.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ[1,##]&@@Abs[DeleteCases[Differences[Append[#,First[#]]],0]]&]],{n,15}]
  • PARI
    step(R,n,D)={matrix(n, n, i, j, if(i>j, sum(k=1, #D, my(s=D[k]); if(j>s && j+s<=n, R[i-j, j-s]))) )}
a(n)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)<=1), t=0); while(R, t+=R[n,k]; R=step(R,n,[0,1,-1])); t)} \\ Andrew Howroyd, Aug 23 2019

Formula

a(n) ~ c * d^n, where d = 1.66820206701846111636107... (see A034297), c = 0.65837031047271348106444... - Vaclav Kotesovec, Sep 21 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 23 2019

A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91
Offset: 1

Views

Author

Gus Wiseman, Mar 23 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The prime indices of 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Links

Crossrefs

For differences instead of quotients we have A325456 (count: A240027).
For multiplicities (prime signature) instead of quotients we have A334965.
The version counting strict divisor chains is A342086.
These partitions are counted by A342498 (strict: A342517, ordered: A342493).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Less@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A342525 Heinz numbers of integer partitions with strictly decreasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98
Offset: 1

Views

Author

Gus Wiseman, Mar 23 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The prime indices of 150 are {1,2,3,3}, with first quotients (2,3/2,1), so 150 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}

Links

Crossrefs

For multiplicities (prime signature) instead of quotients we have A304686.
For differences instead of quotients we have A325457 (count: A320470).
The version counting strict divisor chains is A342086.
These partitions are counted by A342499 (strict: A342518, ordered: A342494).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A056239, A067824, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A325405, A334997, A342530.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Greater@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]
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