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.

Showing 1-10 of 11 results. Next

A318726 Number of integer compositions of n that have only one part or whose consecutive parts are indivisible and the last and first part are also indivisible.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 3, 8, 13, 12, 23, 27, 56, 64, 100, 150, 216, 325, 459, 700, 1007, 1493, 2186, 3203, 4735, 6929, 10243, 14952, 22024, 32366, 47558, 69906, 102634, 150984, 221713, 325919, 478842, 703648, 1034104, 1519432, 2233062, 3281004, 4821791, 7085359
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Examples

			The a(10) = 13 compositions:
  (10)
  (7,3) (3,7) (6,4) (4,6)
  (5,3,2) (5,2,3) (3,5,2) (3,2,5) (2,5,3) (2,3,5)
  (3,2,3,2) (2,3,2,3)
The a(11) = 12 compositions:
  (11)
  (9,2) (2,9) (8,3) (3,8) (7,4) (4,7) (6,5) (5,6)
  (5,2,4) (4,5,2) (2,4,5)

Links

Crossrefs

Cf. A000740, A008965, A167606, A285573, A296302, A303362, A304713, A316476, A318727.

Programs

  • Mathematica
    Table[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,({_,x_,y_,_}/;Divisible[x,y])|({y_,_,x_}/;Divisible[x,y])]&]//Length,{n,20}]
  • PARI
    b(n,k,pred)={my(M=matrix(n,n)); for(n=1, n, M[n,n]=pred(k,n); for(j=1, n-1, M[n,j]=sum(i=1, n-j, if(pred(i,j), M[n-j,i], 0)))); sum(i=1, n, if(pred(i,k), M[n,i], 0))}
a(n)={1 + sum(k=1, n-1, b(n-k, k, (i,j)->i%j<>0))} \\ Andrew Howroyd, Sep 08 2018

Formula

a(n) = A328598(n) + 1. - Gus Wiseman, Nov 04 2019

Extensions

a(21)-a(28) from Robert Price, Sep 08 2018
Terms a(29) and beyond from Andrew Howroyd, Sep 08 2018
Name corrected by Gus Wiseman, Nov 04 2019

A328593 Numbers whose binary indices have no consecutive divisible parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 14, 16, 18, 20, 22, 24, 28, 30, 32, 40, 44, 46, 48, 50, 52, 54, 56, 60, 62, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 104, 108, 110, 112, 114, 116, 118, 120, 124, 126, 128, 132, 134, 144, 146, 148, 150, 152, 156, 158, 160
Offset: 1

Views

Author

Gus Wiseman, Oct 21 2019

Keywords

Comments

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.

Examples

			The sequence of terms together with their binary expansions and binary indices begins:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   4:    100 ~ {3}
   6:    110 ~ {2,3}
   8:   1000 ~ {4}
  12:   1100 ~ {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}
  28:  11100 ~ {3,4,5}
  30:  11110 ~ {2,3,4,5}
  32: 100000 ~ {6}
  40: 101000 ~ {4,6}
  44: 101100 ~ {3,4,6}
  46: 101110 ~ {2,3,4,6}
  48: 110000 ~ {5,6}
  50: 110010 ~ {2,5,6}

Crossrefs

The version for prime indices is A328603.
Numbers with no successive binary indices are A003714.
Partitions with no consecutive divisible parts are A328171.
Compositions without consecutive divisible parts are A328460.
Cf. A000120, A014081, A328187, A328508, A328594, A328595, A328598, A328600, A328608.

Programs

  • Mathematica
    Select[Range[0,100],!MatchQ[Join@@Position[Reverse[IntegerDigits[#,2]],1],{_,x_,y_,_}/;Divisible[y,x]]&]

A328600 Number of necklace compositions of n with no part circularly followed by a divisor.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 3, 5, 5, 7, 10, 18, 20, 29, 40, 58, 78, 111, 156, 218, 304, 429, 604, 859, 1209, 1726, 2423, 3462, 4904, 7000, 9953, 14210, 20270, 28979, 41391, 59253, 84799, 121539, 174162, 249931, 358577, 515090, 739932, 1063826, 1529766, 2201382, 3168565
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2019

Keywords

Comments

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.
Circularity means the last part is followed by the first.

Examples

			The a(5) = 1 through a(13) = 18 necklace compositions (empty column not shown):
  (2,3)  (2,5)  (3,5)  (2,7)    (3,7)      (2,9)    (5,7)      (4,9)
         (3,4)         (4,5)    (4,6)      (3,8)    (2,3,7)    (5,8)
                       (2,4,3)  (2,3,5)    (4,7)    (2,7,3)    (6,7)
                                (2,5,3)    (5,6)    (3,4,5)    (2,11)
                                (2,3,2,3)  (2,4,5)  (3,5,4)    (3,10)
                                                    (2,3,2,5)  (2,4,7)
                                                    (2,3,4,3)  (2,6,5)
                                                               (2,8,3)
                                                               (3,6,4)
                                                               (2,3,5,3)

Links

Crossrefs

The non-necklace version is A328598.
The version with singletons is A318729.
The case forbidding multiples as well as divisors is A328601.
The non-necklace, non-circular version is A328460.
The version for co-primality (instead of divisibility) is A328602.
Necklace compositions are A008965.
Partitions with no part followed by a divisor are A328171.
Cf. A032153, A167606, A318748, A328508, A328593, A328599, A328603, A328608, A328609.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]&]],{n,10}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->i%j<>0))); vector(n, n, sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 26 2019

Formula

a(n) = A318729(n) - 1.

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

A328601 Number of necklace compositions of n with no part circularly followed by a divisor or a multiple.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 2, 5, 4, 7, 6, 13, 14, 20, 30, 38, 50, 68, 97, 132, 176, 253, 328, 470, 631, 901, 1229, 1709, 2369, 3269, 4590, 6383, 8897, 12428, 17251, 24229, 33782, 47404, 66253, 92859, 130141, 182468, 256261, 359675, 505230, 710058, 997952, 1404214
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2019

Keywords

Comments

A necklace composition of n (A008965) is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.
Circularity means the last part is followed by the first.

Examples

			The a(5) = 1 through a(13) = 6 necklace compositions (empty column not shown):
  (2,3)  (2,5)  (3,5)  (2,7)  (3,7)      (2,9)  (5,7)      (4,9)
         (3,4)         (4,5)  (4,6)      (3,8)  (2,3,7)    (5,8)
                              (2,3,5)    (4,7)  (2,7,3)    (6,7)
                              (2,5,3)    (5,6)  (3,4,5)    (2,11)
                              (2,3,2,3)         (3,5,4)    (3,10)
                                                (2,3,2,5)  (2,3,5,3)
                                                (2,3,4,3)

Links

Crossrefs

The non-necklace version is A328599.
The case forbidding divisors only is A328600 or A318729 (with singletons).
The non-necklace, non-circular version is A328508.
The version for co-primality (instead of indivisibility) is A328597.
Cf. A000740, A008965, A032153, A167606, A318748, A328171, A328460, A328593, A328598, A328602, A328603, A328608, A328609.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]&&And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]&]],{n,10}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->i%j<>0 && j%i<>0))); vector(n, n, sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 26 2019

Formula

a(n) = A318730(n) - 1.

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

A328609 Number of compositions of n whose circularly adjacent parts are relatively prime.

Original entry on oeis.org

1, 1, 1, 3, 6, 12, 23, 42, 81, 150, 284, 534, 1004, 1882, 3532, 6630, 12459, 23406, 43951, 82537, 154998, 291087, 546673, 1026686, 1928117, 3621016, 6800299, 12771085, 23984328, 45042958, 84591338, 158863806, 298348612, 560303341, 1052258401, 1976157509
Offset: 0

Views

Author

Gus Wiseman, Oct 26 2019

Keywords

Comments

Circularity means the last part is followed by the first.

Examples

			The a(1) = 1 through a(6) = 23 compositions:
  (1)  (11)  (12)   (13)    (14)     (15)
             (21)   (31)    (23)     (51)
             (111)  (112)   (32)     (114)
                    (121)   (41)     (123)
                    (211)   (113)    (132)
                    (1111)  (131)    (141)
                            (311)    (213)
                            (1112)   (231)
                            (1121)   (312)
                            (1211)   (321)
                            (2111)   (411)
                            (11111)  (1113)
                                     (1131)
                                     (1212)
                                     (1311)
                                     (2121)
                                     (3111)
                                     (11112)
                                     (11121)
                                     (11211)
                                     (12111)
                                     (21111)
                                     (111111)

Links

Crossrefs

The necklace version is A328597 or A318728 (with singletons).
The aperiodic version is A328670.
The Lyndon word version is A318745.
The version with singletons is A318748.
The non-circular version is A167606.
Relatively prime compositions are A000740.
Compositions with no part circularly followed by a divisor are A328598.
Cf. A008965, A178470, A318726, A325680, A328172, A328335, A328599, A328600.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@CoprimeQ@@@Partition[#,2,1,1]&]],{n,10}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i, j)->gcd(i, j)==1)))} \\ Andrew Howroyd, Nov 01 2019

Formula

a(n > 1) = A318748(n) - 1.

A328603 Numbers whose prime indices have no consecutive divisible parts, meaning no prime index is a divisor of the next-smallest prime index, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167
Offset: 1

Views

Author

Gus Wiseman, Oct 26 2019

Keywords

Comments

First differs from A304713 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.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   31: {11}
   33: {2,5}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   51: {2,7}

Crossrefs

A subset of A005117.
These are the Heinz numbers of the partitions counted by A328171.
The non-strict version is A328674 (Heinz numbers for A328675).
The version for relatively prime instead of indivisible is A328335.
Compositions without consecutive divisibilities are A328460.
Numbers whose binary indices lack consecutive divisibilities are A328593.
The version with all pairs indivisible is A304713.
Cf. A056239, A112798, A316476, A318726, A318729, A326704, A328336, A328598, A328599.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MatchQ[primeMS[#],{_,x_,y_,_}/;Divisible[y,x]]&]

Formula

Intersection of A005117 and A328674.

A328597 Number of necklace compositions of n where every pair of adjacent parts (including the last with the first) is relatively prime.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 21, 33, 57, 94, 167, 279, 491, 852, 1507, 2647, 4714, 8349, 14923, 26642, 47793, 85778, 154474, 278322, 502715, 908912, 1646205, 2984546, 5418652, 9847189, 17916000, 32625617, 59470539, 108493149, 198094482, 361965238, 661891579, 1211162270
Offset: 1

Views

Author

Gus Wiseman, Oct 23 2019

Keywords

Comments

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

Examples

			The a(1) = 1 through a(7) = 12 necklace compositions:
  (1)  (1,1)  (1,2)    (1,3)      (1,4)        (1,5)          (1,6)
              (1,1,1)  (1,1,2)    (2,3)        (1,1,4)        (2,5)
                       (1,1,1,1)  (1,1,3)      (1,2,3)        (3,4)
                                  (1,1,1,2)    (1,3,2)        (1,1,5)
                                  (1,1,1,1,1)  (1,1,1,3)      (1,1,1,4)
                                               (1,2,1,2)      (1,1,2,3)
                                               (1,1,1,1,2)    (1,1,3,2)
                                               (1,1,1,1,1,1)  (1,2,1,3)
                                                              (1,1,1,1,3)
                                                              (1,1,2,1,2)
                                                              (1,1,1,1,1,2)
                                                              (1,1,1,1,1,1,1)

Links

Crossrefs

The non-necklace version is A328609.
The non-necklace non-circular version is A167606.
The version with singletons is A318728.
The aperiodic case is A318745.
The indivisible (instead of coprime) version is A328600.
The non-coprime (instead of coprime) version is A328602.
Necklace compositions are A008965.
Cf. A000031, A000740, A000837, A032153, A059966, A318729, A318748, A328172, A328598, A328599, A328601.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]&]],{n,10}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->gcd(i,j)==1))); vector(n, n, sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 26 2019

Formula

a(n > 1) = A318728(n) - 1.

Extensions

Terms a(21) and beyond from Andrew Howroyd, Oct 26 2019

A328599 Number of compositions of n with no part circularly followed by a divisor or a multiple.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 0, 4, 2, 4, 12, 8, 22, 14, 36, 44, 62, 114, 130, 206, 264, 414, 602, 822, 1250, 1672, 2520, 3518, 5146, 7408, 10448, 15224, 21496, 31284, 44718, 64170, 92314, 131618, 190084, 271870, 391188, 560978, 804264, 1155976, 1656428, 2381306, 3414846
Offset: 0

Views

Author

Gus Wiseman, Oct 25 2019

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
Circularity means the last part is followed by the first.

Examples

			The a(0) = 1 through a(12) = 22 compositions (empty columns not shown):
  ()  (2,3)  (2,5)  (3,5)  (2,7)  (3,7)      (2,9)  (5,7)
      (3,2)  (3,4)  (5,3)  (4,5)  (4,6)      (3,8)  (7,5)
             (4,3)         (5,4)  (6,4)      (4,7)  (2,3,7)
             (5,2)         (7,2)  (7,3)      (5,6)  (2,7,3)
                                  (2,3,5)    (6,5)  (3,2,7)
                                  (2,5,3)    (7,4)  (3,4,5)
                                  (3,2,5)    (8,3)  (3,5,4)
                                  (3,5,2)    (9,2)  (3,7,2)
                                  (5,2,3)           (4,3,5)
                                  (5,3,2)           (4,5,3)
                                  (2,3,2,3)         (5,3,4)
                                  (3,2,3,2)         (5,4,3)
                                                    (7,2,3)
                                                    (7,3,2)
                                                    (2,3,2,5)
                                                    (2,3,4,3)
                                                    (2,5,2,3)
                                                    (3,2,3,4)
                                                    (3,2,5,2)
                                                    (3,4,3,2)
                                                    (4,3,2,3)
                                                    (5,2,3,2)

Links

Crossrefs

The necklace version is A328601.
The case forbidding only divisors (not multiples) is A328598.
The non-circular version is A328508.
Partitions with no part followed by a divisor are A328171.
Cf. A000740, A008965, A167606, A318729, A318748, A328460, A328593, A328600, A328603, A328608, A328609, A328674.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Not/@Divisible@@@Partition[#,2,1,1]&&And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]&]],{n,0,10}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i,j)->i%j<>0&&j%i<>0)))} \\ Andrew Howroyd, Oct 26 2019

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Oct 25 2019

Keywords

Comments

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.

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Oct 29 2019

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

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