A329139 -id:A329139 - cp's OEIS frontend

A329132 Numbers whose augmented differences of prime indices are a periodic sequence.

4, 8, 15, 16, 32, 55, 64, 90, 105, 119, 128, 225, 253, 256, 403, 512, 540, 550, 697, 893, 935, 1024, 1155, 1350, 1357, 1666, 1943, 2048, 2263, 3025, 3071, 3150, 3240, 3375, 3451, 3927, 3977, 4096, 4429, 5123, 5500, 5566, 6731, 7735, 8083, 8100, 8192, 9089

Offset: 1

Gus Wiseman, Nov 06 2019

nonn

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 sequence is periodic if its cyclic rotations are not all different.

			The sequence of terms together with their augmented differences of prime indices begins:
     4: (1,1)
     8: (1,1,1)
    15: (2,2)
    16: (1,1,1,1)
    32: (1,1,1,1,1)
    55: (3,3)
    64: (1,1,1,1,1,1)
    90: (2,1,2,1)
   105: (2,2,2)
   119: (4,4)
   128: (1,1,1,1,1,1,1)
   225: (1,2,1,2)
   253: (5,5)
   256: (1,1,1,1,1,1,1,1)
   403: (6,6)
   512: (1,1,1,1,1,1,1,1,1)
   540: (2,1,1,2,1,1)
   550: (3,1,3,1)
   697: (7,7)
   893: (8,8)

Complement of A329133.
These are the Heinz numbers of the partitions counted by A329143.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Numbers whose differences of prime indices are periodic are A329134.
Cf. A000961, A027375, A056239, A112798, A325356, A325389, A325394, A328594, A329135, A329136, A329139.

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

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

Gus Wiseman, Nov 09 2019

nonn

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.

			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)

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.

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

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

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

Gus Wiseman, Nov 09 2019

nonn

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.

			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)

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.

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

A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 2, 4, 6, 4, 0, 0, 4, 6, 9, 8, 5, 0, 0, 2, 6, 15, 20, 15, 6, 0, 0, 4, 8, 24, 32, 35, 18, 7, 0, 0, 3, 10, 27, 56, 70, 54, 28, 8, 0, 0, 4, 12, 42, 84, 125, 120, 84, 32, 9, 0, 0, 2, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Gus Wiseman, Apr 16 2020

A composition of n is a finite sequence of positive integers summing to n.

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   3   2   3   0
   0   2   4   6   4   0
   0   4   6   9   8   5   0
   0   2   6  15  20  15   6   0
   0   4   8  24  32  35  18   7   0
   0   3  10  27  56  70  54  28   8   0
   0   4  12  42  84 125 120  84  32   9   0
   0   2  10  45 120 210 252 210 120  45  10   0
   0   6  18  66 168 335 450 462 320 162  50  11   0
Row n = 6 counts the following compositions (empty columns indicated by dots):
  .  (6)       (15)    (114)  (1113)  (11112)  .
     (33)      (24)    (123)  (1122)  (11121)
     (222)     (42)    (132)  (1131)  (11211)
     (111111)  (51)    (141)  (1221)  (12111)
               (1212)  (213)  (1311)  (21111)
               (2121)  (231)  (2112)
                       (312)  (2211)
                       (321)  (3111)
                       (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column k = 1 is A000005.
Row sums are A011782.
Diagonal T(2n,n) is A045630(n).
The strict version is A072574.
A version counting runs is A238279.
Column k = n - 1 is A254667.
Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Period of binary expansion is A302291.
Numbers whose prime signature is aperiodic are A329139.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Rotational symmetries are counted by A138904.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A211100, A291166, A328595, A328596, A329312, A329313, A329326.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[c,Length[Union[Array[RotateRight[c,#]&,Length[c]]]]==k]]],{n,0,10},{k,0,n}]

T(n,k)=if(n==0, k==0, sumdiv(n, m, sumdiv(gcd(k,m), d, moebius(d)*binomial(m/d-1, k/d-1)))) \\ Andrew Howroyd, Jan 19 2023

T(n,k) = Sum_{m|n} Sum_{d|gcd(k,m)} mu(d)*binomial(m/d-1, k/d-1) for n > 0. - Andrew Howroyd, Jan 19 2023

A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 6, 1, 5, 3, 3, 1, 9, 2, 3, 4, 5, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 9, 1, 7, 1, 5, 5, 3, 1, 17, 2, 6, 3, 5, 1, 12, 3, 9, 3, 3, 1, 11, 1, 3, 5, 32, 3, 7, 1, 5, 3, 7, 1, 18, 1, 3, 6, 5, 3, 7, 1, 17, 8, 3, 1, 11

Offset: 1

Gus Wiseman, Apr 17 2020

nonn

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

The k-th composition in standard order (row k of 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 unsorted prime signature of 12345678 is (1,2,1,1), which is the 27th composition in standard order, so a(12345678) = 27.

Positions of first appearances are A057335 (a partial inverse).
Least number with same prime signature is A071364.
Unsorted prime signature is A124010.
Least number with reversed prime signature is A331580.
Minimal numbers with standard reversed prime signatures are A334031.
The reversed version is A334033.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A029931, A048793, A052409, A055932, A056239, A112798, A124767, A228351, A233249, A329139, A333220.

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Last/@If[n==1,{},FactorInteger[n]]],{n,100}]

a(A057335(n)) = n.
A057335(a(n)) = A071364(n).
a(A334031(n))= A059893(n).
A334031(a(n)) = A331580(n).

A329137 Number of integer partitions of n whose differences are an aperiodic word.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 14, 20, 25, 39, 54, 69, 99, 130, 167, 224, 292, 373, 483, 620, 773, 993, 1246, 1554, 1946, 2421, 2987, 3700, 4548, 5575, 6821, 8330, 10101, 12287, 14852, 17935, 21599, 25986, 31132, 37295, 44539, 53112, 63212, 75123, 89055, 105503, 124682

Offset: 0

Gus Wiseman, Nov 09 2019

nonn

A sequence is aperiodic if its cyclic rotations are all different.

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)    (3)    (4)      (5)        (6)          (7)
       (1,1)  (2,1)  (2,2)    (3,2)      (3,3)        (4,3)
                     (3,1)    (4,1)      (4,2)        (5,2)
                     (2,1,1)  (2,2,1)    (5,1)        (6,1)
                              (3,1,1)    (4,1,1)      (3,2,2)
                              (2,1,1,1)  (2,2,1,1)    (3,3,1)
                                         (3,1,1,1)    (4,2,1)
                                         (2,1,1,1,1)  (5,1,1)
                                                      (2,2,2,1)
                                                      (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 differences:
  ()  ()   ()   ()     ()       ()         ()
      (0)  (1)  (0)    (1)      (0)        (1)
                (2)    (3)      (2)        (3)
                (1,0)  (0,1)    (4)        (5)
                       (2,0)    (3,0)      (0,2)
                       (1,0,0)  (0,1,0)    (1,0)
                                (2,0,0)    (2,1)
                                (1,0,0,0)  (4,0)
                                           (0,0,1)
                                           (1,1,0)
                                           (3,0,0)
                                           (0,1,0,0)
                                           (2,0,0,0)
                                           (1,0,0,0,0)

The Heinz numbers of these partitions are given by A329135.
The periodic version is A329144.
The augmented version is 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.
Cf. A152061, A325356, A329132, A329133, A329134, A329140.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Table[Length[Select[IntegerPartitions[n],aperQ[Differences[#]]&]],{n,0,30}]

a(n) + A329144(n) = A000041(n).

A331580 Smallest number whose unsorted prime signature is the reversed unsorted prime signature of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 18, 2, 6, 6, 16, 2, 12, 2, 18, 6, 6, 2, 54, 4, 6, 8, 18, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 54, 2, 30, 2, 18, 18, 6, 2, 162, 4, 12, 6, 18, 2, 24, 6, 54, 6, 6, 2, 150, 2, 6, 18, 64, 6, 30, 2, 18, 6, 30, 2, 108, 2, 6, 12, 18, 6

Offset: 1

Gus Wiseman, Apr 18 2020

nonn

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

			The prime signature of 12345678 = 2*3*3*47*14593 is (1,2,1,1), and the least number with prime signature (1,1,2,1) is 1050 = 2*3*5*5*7, so a(12345678) = 1050.

The range is A055932.
The non-reversed version is A071364.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A233249, A333217, A333219, A334032, A334033.

ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
Table[Times@@Prime/@ptnToNorm[Reverse[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

A334031 The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.

Original entry on oeis.org

1, 2, 4, 6, 8, 18, 12, 30, 16, 54, 36, 150, 24, 90, 60, 210, 32, 162, 108, 750, 72, 450, 300, 1470, 48, 270, 180, 1050, 120, 630, 420, 2310, 64, 486, 324, 3750, 216, 2250, 1500, 10290, 144, 1350, 900, 7350, 600, 4410, 2940, 25410, 96, 810, 540, 5250, 360, 3150

Offset: 0

Gus Wiseman, Apr 17 2020

nonn

All terms are normal (A055932), meaning their prime indices cover an initial interval of positive integers.
Unsorted prime signature is the sequence of exponents in a number's prime factorization.
The k-th composition in standard order (row k of 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 sequence of terms together with their prime indices begins:
       1: {}
       2: {1}
       4: {1,1}
       6: {1,2}
       8: {1,1,1}
      18: {1,2,2}
      12: {1,1,2}
      30: {1,2,3}
      16: {1,1,1,1}
      54: {1,2,2,2}
      36: {1,1,2,2}
     150: {1,2,3,3}
      24: {1,1,1,2}
      90: {1,2,2,3}
      60: {1,1,2,3}
     210: {1,2,3,4}
      32: {1,1,1,1,1}
     162: {1,2,2,2,2}
For example, the 13th composition in standard order is (1,2,1), and the least number with prime signature (1,2,1) is 90 = 2^1 * 3^2 * 5^1, so a(13) = 90.

The range is A055932.
The non-reversed version is A057335.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Normal numbers with standard compositions as prime signature are A334032.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Heinz number is A333219.
Cf. A029931, A048793, A056239, A112798, A228351, A233249, A272020, A333218, A333220, A334032.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Product[Prime[i]^stc[n][[-i]],{i,DigitCount[n,2,1]}],{n,0,100}]

a(n) = A057335(A059893(n)).

A334033 The a(n)-th composition in standard order (graded reverse-lexicographic) is the reversed unsorted prime signature of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 6, 1, 3, 3, 8, 1, 5, 1, 6, 3, 3, 1, 12, 2, 3, 4, 6, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 12, 1, 7, 1, 6, 6, 3, 1, 24, 2, 5, 3, 6, 1, 9, 3, 12, 3, 3, 1, 14, 1, 3, 6, 32, 3, 7, 1, 6, 3, 7, 1, 20, 1, 3, 5, 6, 3, 7, 1, 24, 8, 3, 1

Offset: 1

Gus Wiseman, Apr 18 2020

nonn

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

The k-th composition in standard order (row k of 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 unsorted prime signature of 12345678 is (1,2,1,1), whose reverse (1,1,2,1) is the 29th composition in standard order, so a(12345678) = 29.

Positions of first appearances are A334031.
The non-reversed version is A334032.
Unsorted prime signature is A124010.
Least number with reversed prime signature is A331580.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A029931, A048793, A052409, A055932, A056239, A057335, A071364, A112798, A124767, A228351, A233249, A329139, A333220.

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Reverse[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

a(A334031(n)) = n.
A334031(a(n)) = A071364(n).
a(A057335(n))= A059893(n).
A057335(a(n)) = A331580(n).

A334298 Numbers whose prime signature is a reversed Lyndon word.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116

Offset: 1

Gus Wiseman, Jun 10 2020

nonn

A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.

A number's prime signature is the sequence of positive exponents in its prime factorization.

			The prime signature of 4200 is (3,1,2,1), which is a reversed Lyndon word, so 4200 is in the sequence.
The sequence of terms together with their prime indices begins:
   1: {}           23: {9}            48: {1,1,1,1,2}
   2: {1}          24: {1,1,1,2}      49: {4,4}
   3: {2}          25: {3,3}          52: {1,1,6}
   4: {1,1}        27: {2,2,2}        53: {16}
   5: {3}          28: {1,1,4}        56: {1,1,1,4}
   7: {4}          29: {10}           59: {17}
   8: {1,1,1}      31: {11}           60: {1,1,2,3}
   9: {2,2}        32: {1,1,1,1,1}    61: {18}
  11: {5}          37: {12}           63: {2,2,4}
  12: {1,1,2}      40: {1,1,1,3}      64: {1,1,1,1,1,1}
  13: {6}          41: {13}           67: {19}
  16: {1,1,1,1}    43: {14}           68: {1,1,7}
  17: {7}          44: {1,1,5}        71: {20}
  19: {8}          45: {2,2,3}        72: {1,1,1,2,2}
  20: {1,1,3}      47: {15}           73: {21}

The non-reversed version is A329131.
Lyndon compositions are A059966.
Prime signature is A124010.
Numbers with strictly decreasing prime multiplicities are A304686.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose prime signature is a necklace are A329138.
Numbers whose prime signature is aperiodic are A329139.
Cf. A000031, A000740, A000961, A001037, A025487, A027375, A097318, A112798, A118914, A304678, A318731, A329140, A329142.

lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
Select[Range[100],lynQ[Reverse[Last/@If[#==1,{},FactorInteger[#]]]]&]

cp's OEIS Frontend

A329132 Numbers whose augmented differences of prime indices are a periodic sequence.

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

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

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A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

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A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.

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A329137 Number of integer partitions of n whose differences are an aperiodic word.

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A331580 Smallest number whose unsorted prime signature is the reversed unsorted prime signature of n.

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A334031 The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.

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