A316476 -id:A316476 - cp's OEIS frontend

A328677 Numbers whose distinct prime indices are relatively prime and pairwise indivisible.

2, 4, 8, 15, 16, 32, 33, 35, 45, 51, 55, 64, 69, 75, 77, 85, 93, 95, 99, 119, 123, 128, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 205, 207, 209, 215, 217, 219, 221, 225, 245, 249, 253, 255, 256, 265, 275, 279, 287, 291, 295, 297, 309, 323

Offset: 1

Gus Wiseman, Oct 30 2019

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. Stable numbers are listed in A316476.

			The sequence of terms together with their prime indices begins:
    2: {1}
    4: {1,1}
    8: {1,1,1}
   15: {2,3}
   16: {1,1,1,1}
   32: {1,1,1,1,1}
   33: {2,5}
   35: {3,4}
   45: {2,2,3}
   51: {2,7}
   55: {3,5}
   64: {1,1,1,1,1,1}
   69: {2,9}
   75: {2,3,3}
   77: {4,5}
   85: {3,7}
   93: {2,11}
   95: {3,8}
   99: {2,2,5}
  119: {4,7}

These are the Heinz numbers of the partitions counted by A328676.
Numbers whose prime indices are relatively prime are A289509.
Partitions whose distinct parts are pairwise indivisible are A305148.
The version for binary indices (instead of prime indices) is A328671.
Cf. A000837, A056239, A112798, A285573, A289508, A303362, A304713, A327393, A328460.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],GCD@@primeMS[#]==1&&stableQ[primeMS[#],Divisible]&]

Intersection of A316476 and A289509.

A329557 Smallest MM-number of a set of n nonempty sets.

Original entry on oeis.org

1, 3, 15, 165, 2145, 36465, 1057485, 32782035, 1344063435, 57794727705, 2716352202135, 160264779925965, 10737740255039655, 783855038617894815, 61924548050813690385, 5139737488217536301955, 519113486309971166497455, 56583370007786857148222595, 6393920810879914857749153235

Offset: 0

Gus Wiseman, Nov 17 2019

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
        1: {}
        3: {{1}}
       15: {{1},{2}}
      165: {{1},{2},{3}}
     2145: {{1},{2},{3},{1,2}}
    36465: {{1},{2},{3},{1,2},{4}}
  1057485: {{1},{2},{3},{1,2},{4},{1,3}}

MM-numbers of sets of sets are A302494.
MM-numbers of sets of nonempty sets are A329629.
The version allowing empty sets is A329558.
The version without singletons is A329554.
Cf. A056239, A072639, A112798, A302242, A326031, A329552, A329555, A329556.
Other MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dae=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],1]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

a(n) = my(k=1); prod(i=1, n, until(issquarefree(k), k++); prime(k)); \\ Jinyuan Wang, Feb 23 2025

a(n) = A329558(n + 1)/2.

More terms from Jinyuan Wang, Feb 23 2025

A318727 Number of integer compositions of n where adjacent parts are indivisible (either way) and the last and first part are also indivisible (either way).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 3, 5, 13, 9, 23, 15, 37, 45, 63, 115, 131, 207, 265, 415, 603, 823, 1251, 1673, 2521, 3519, 5147, 7409, 10449, 15225, 21497, 31285, 44719, 64171, 92315, 131619, 190085, 271871, 391189, 560979, 804265, 1155977, 1656429, 2381307, 3414847

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			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)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000740, A008965, A167606, A285573, A296302, A303362, A304713, A316476, A318726.

Table[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,({_,x_,y_,_}/;Divisible[x,y]||Divisible[y,x])|({y_,_,x_}/;Divisible[x,y]||Divisible[y,x])]&]//Length,{n,20}]

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&&j%i<>0))} \\ Andrew Howroyd, Sep 08 2018

a(21)-a(28) from Robert Price, Sep 07 2018

Terms a(29) and beyond from Andrew Howroyd, Sep 08 2018

A328671 Numbers whose binary indices are relatively prime and pairwise indivisible.

Original entry on oeis.org

1, 6, 12, 18, 20, 22, 24, 28, 48, 56, 66, 68, 70, 72, 76, 80, 82, 84, 86, 88, 92, 96, 104, 112, 120, 132, 144, 148, 176, 192, 196, 208, 212, 224, 240, 258, 264, 272, 274, 280, 296, 304, 312, 320, 322, 328, 336, 338, 344, 352, 360, 368, 376, 384, 400, 416, 432

Offset: 1

Gus Wiseman, Oct 29 2019

nonn

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.

			The sequence of terms together with their binary expansions and binary indices begins:
    1:         1 ~ {1}
    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}
   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}
   80:   1010000 ~ {5,7}
   82:   1010010 ~ {2,5,7}
   84:   1010100 ~ {3,5,7}
   86:   1010110 ~ {2,3,5,7}
   88:   1011000 ~ {4,5,7}

The version for prime indices (instead of binary indices) is A328677.
Numbers whose binary indices are relatively prime are A291166.
Numbers whose distinct prime indices are pairwise indivisible are A316476.
BII-numbers of antichains are A326704.
Relatively prime partitions whose distinct parts are pairwise indivisible are A328676, with strict case A328678.
Cf. A000120, A121016, A285573, A303362, A304713, A305148, A316468, A316475.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],GCD@@bpe[#]==1&&stableQ[bpe[#],Divisible]&]

Intersection of A291166 with A326704.

A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 40, 64, 128, 160, 256, 288, 296, 416, 512, 520, 544, 552, 640, 672, 800, 808, 928, 1024, 2048, 2176, 2304, 2432, 2560, 2688, 2816, 2944, 4096, 8192, 8200, 8224, 8232, 8320, 8352, 8480, 8488, 8608, 8704, 8712, 8736, 8744, 8832, 8864, 8992

Offset: 1

Gus Wiseman, Mar 28 2024

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 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.

			The terms together with their prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    4: {{2}}
    8: {{1,1}}
   16: {{3}}
   32: {{1,2}}
   40: {{1,1},{1,2}}
   64: {{4}}
  128: {{1,1,1}}
  160: {{1,2},{1,1,1}}
  256: {{2,2}}
  288: {{1,2},{2,2}}
  296: {{1,1},{1,2},{2,2}}
  416: {{1,2},{1,1,1},{2,2}}
  512: {{1,3}}
  520: {{1,1},{1,3}}
  544: {{1,2},{1,3}}
  552: {{1,1},{1,2},{1,3}}
  640: {{1,1,1},{1,3}}
  672: {{1,2},{1,1,1},{1,3}}
  800: {{1,2},{2,2},{1,3}}
  808: {{1,1},{1,2},{2,2},{1,3}}
  928: {{1,2},{1,1,1},{2,2},{1,3}}

Connected case of A087086, relatively prime A328671.
For binary indices of binary indices we have A326750, non-primitive A326749.
For prime indices of prime indices we have A329559, non-primitive A305078.
Primitive case of A371291 = positions of ones in A371452.
For binary indices of prime indices we have A371445, non-primitive A325118.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326964 counts connected set-systems, covering A323818.
Cf. A001222, A051026, A285572, A303362, A304713, A305079, A316476, A319496, A319719, A326704, A371446.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],stableQ[bpe[#],Divisible]&&connectedQ[prix/@bpe[#]]&]

Intersection of A087086 and A371291.

A327393 Maximum stable divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 15, 16, 17, 9, 19, 5, 7, 11, 23, 8, 25, 13, 27, 7, 29, 15, 31, 32, 33, 17, 35, 9, 37, 19, 13, 8, 41, 7, 43, 11, 45, 23, 47, 16, 49, 25, 51, 13, 53, 27, 55, 8, 19, 29, 59, 15, 61, 31, 9, 64, 13, 33, 67, 17, 69, 35, 71

Offset: 1

Gus Wiseman, Sep 15 2019

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 number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476, which is the union of this sequence without 1.

			The stable divisors of 60 are {1, 2, 3, 4, 5, 15}, so a(60) = 15.

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A000005, A006530, A302242, A303362.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Max[Select[Divisors[n],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]],{n,100}]

A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 4, 1, 1, 2, 7, 1, 6, 1, 3, 3, 10, 1, 9, 3, 5, 4, 17, 1, 23, 6, 7, 6, 20, 3, 36, 9, 15, 7, 45, 5, 56, 14, 17, 20, 65, 7, 83, 18, 40

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

Positions of terms greater than 1 are {31, 37, 41, 43, 46, 47, 49, ...}.

A partition with no two distinct parts relatively prime is said to be intersecting.

			Examples:
  a(31) = 2:         a(46) = 2:
    (15,10,6)          (15,15,10,6)
    (1^31)             (1^46)
  a(37) = 3:         a(47) = 7:
    (15,12,10)         (20,15,12)
    (15,10,6,6)        (21,14,12)
    (1^37)             (20,15,6,6)
  a(41) = 4:           (21,14,6,6)
    (20,15,6)          (15,12,10,10)
    (21,14,6)          (15,10,10,6,6)
    (15,10,10,6)       (1^47)
    (1^41)           a(49) = 6:
  a(43) = 4:           (24,15,10)
    (18,15,10)         (18,15,10,6)
    (15,12,10,6)       (15,12,12,10)
    (15,10,6,6,6)      (15,12,10,6,6)
    (1^43)             (15,10,6,6,6,6)
                       (1^39)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400

The Heinz numbers of these partitions are A328679.
The strict case is A318715.
The version for non-isomorphic multiset partitions is A319759.
Relatively prime partitions are A000837.
Intersecting partitions are A328673.
Cf. A078374, A285573, A289509, A291166, A303140, A305148, A316476, A326910, A326912.

Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]]&]],{n,0,32}]

a(n > 0) = A202425(n) + 1.

A329552 Smallest MM-number of a connected set of n sets.

Original entry on oeis.org

1, 2, 39, 195, 5655, 62205, 2674815

Offset: 0

Gus Wiseman, Nov 17 2019

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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
        1: {}
        2: {{}}
       39: {{1},{1,2}}
      195: {{1},{2},{1,2}}
     5655: {{1},{2},{1,2},{1,3}}
    62205: {{1},{2},{3},{1,2},{1,3}}
  2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}

MM-numbers of connected set-systems are A328514.
The weight of the system with MM-number n is A302242(n).
Connected numbers are A305078.
Maximum connected divisor is A327076.
BII-numbers of connected sets of sets are A326749.
The smallest BII-number of a connected set of n sets is A329625(n).
Allowing edges to have repeated vertices gives A329553.
Requiring the edges to form an antichain gives A329555.
The smallest MM-number of a set of n nonempty sets is A329557(n).
Cf. A048143, A056239, A112798, A302494, A304714, A304716, A305079, A322389, A328513, A329554, A329556, A329558.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
da=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[da[[Position[PrimeOmega/@da,n][[1,1]]]],{n,First[Split[Union[PrimeOmega/@da],#2==#1+1&]]}]

A329555 Smallest MM-number of a clutter (connected antichain) of n distinct sets.

Original entry on oeis.org

1, 2, 377, 16211, 761917

Offset: 0

Gus Wiseman, Nov 17 2019

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
       1: {}
       2: {{}}
     377: {{1,2},{1,3}}
   16211: {{1,2},{1,3},{1,4}}
  761917: {{1,2},{1,3},{1,4},{2,3}}

Spanning cutters of distinct sets are counted by A048143.
MM-numbers of connected weak-antichains are A329559.
MM-numbers of sets of sets are A302494.
The smallest BII-number of a clutter with n edges is A329627.
Not requiring the edges to form an antichain gives A329552.
Connected numbers are A305078.
Stable numbers are A316476.
Cf. A056239, A112798, A302242, A319837, A320275, A322113, A327076, A328514, A329552, A329558, A329560, A329561.
Other MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&&stableQ[primeMS[#],Divisible]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

A329558 Product of primes indexed by the first n squarefree numbers.

Original entry on oeis.org

1, 2, 6, 30, 330, 4290, 72930, 2114970, 65564070, 2688126870, 115589455410, 5432704404270, 320529559851930, 21475480510079310, 1567710077235789630, 123849096101627380770, 10279474976435072603910, 1038226972619942332994910, 113166740015573714296445190, 12787841621759829715498306470

Offset: 0

Gus Wiseman, Nov 17 2019

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}. Then a(n) is the smallest MM-number of a set of n sets.

			The sequence of terms together with their corresponding systems begins:
        1: {}
        2: {{}}
        6: {{},{1}}
       30: {{},{1},{2}}
      330: {{},{1},{2},{3}}
     4290: {{},{1},{2},{3},{1,2}}
    72930: {{},{1},{2},{3},{1,2},{4}}
  2114970: {{},{1},{2},{3},{1,2},{4},{1,3}}

Jinyuan Wang, Table of n, a(n) for n = 0..100

The smallest BII-number of a set of n sets is A000225(n).
MM-numbers of sets of sets are A302494.
The case without empty edges is A329557.
The case without singletons is A329556.
The case without empty edges or singletons is A329554.
The connected version is A329552.
Cf. A005117, A048672, A056239, A072639, A112798, A302242, A326031, A329555.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

sqvs=Select[Range[30],SquareFreeQ];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]

a(n > 0) = 2 * A329557(n - 1).

a(n) = Product_{i = 1..n} prime(A005117(i)).

a(19) from Jinyuan Wang, Feb 24 2020

cp's OEIS Frontend

A328677 Numbers whose distinct prime indices are relatively prime and pairwise indivisible.

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A329557 Smallest MM-number of a set of n nonempty sets.

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A318727 Number of integer compositions of n where adjacent parts are indivisible (either way) and the last and first part are also indivisible (either way).

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A328671 Numbers whose binary indices are relatively prime and pairwise indivisible.

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A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

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A327393 Maximum stable divisor of n.

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A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

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A329552 Smallest MM-number of a connected set of n sets.

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