A302569 -id:A302569 - cp's OEIS frontend

A340608 The number of prime factors of n (A001222) is relatively prime to the maximum prime index of n (A061395).

2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 22, 23, 25, 27, 28, 29, 31, 32, 33, 34, 37, 40, 41, 42, 43, 44, 46, 47, 48, 51, 53, 55, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 76, 77, 79, 80, 82, 83, 85, 88, 89, 90, 93, 94, 97, 98, 99

Offset: 1

Gus Wiseman, Jan 27 2021

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 sequence of terms together with their prime indices begins:
     2: {1}          22: {1,5}          44: {1,1,5}
     3: {2}          23: {9}            46: {1,9}
     4: {1,1}        25: {3,3}          47: {15}
     5: {3}          27: {2,2,2}        48: {1,1,1,1,2}
     7: {4}          28: {1,1,4}        51: {2,7}
     8: {1,1,1}      29: {10}           53: {16}
    10: {1,3}        31: {11}           55: {3,5}
    11: {5}          32: {1,1,1,1,1}    59: {17}
    12: {1,1,2}      33: {2,5}          60: {1,1,2,3}
    13: {6}          34: {1,7}          61: {18}
    15: {2,3}        37: {12}           62: {1,11}
    16: {1,1,1,1}    40: {1,1,1,3}      63: {2,2,4}
    17: {7}          41: {13}           64: {1,1,1,1,1,1}
    18: {1,2,2}      42: {1,2,4}        66: {1,2,5}
    19: {8}          43: {14}           67: {19}

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

Note: Heinz numbers are given in parentheses below.
These are the Heinz numbers of the partitions counted by A200750.
The case of equality is A047993 (A106529).
The divisible instead of coprime version is A168659 (A340609).
The dividing instead of coprime version is A168659 (A340610), with strict case A340828 (A340856).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A051424 counts singleton or pairwise coprime partitions (A302569).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A259936 counts singleton or pairwise coprime factorizations.
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
A326849 counts partitions whose sum divides length times maximum (A326848).
A327516 counts pairwise coprime partitions (A302696).
Cf. A003114, A039900, A096401, A244990/A244991, A340606, A340653, A340691, A340787/A340788.

Select[Range[100],GCD[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==1&]

A303140 Number of strict integer partitions of n with at least two but not all parts having a common divisor greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 4, 2, 8, 7, 14, 14, 21, 18, 33, 32, 50, 54, 72, 67, 103, 110, 145, 155, 201, 196, 271, 293, 372, 400, 493, 512, 647, 704, 858, 924, 1115, 1167, 1436, 1560, 1854, 2022, 2368, 2510, 3005, 3255, 3804, 4144, 4792, 5116, 5989, 6514, 7486

Offset: 1

Gus Wiseman, Apr 19 2018

nonn

			The a(14) = 7 partitions are (932), (8321), (7421), (653), (6521), (6431), (5432).

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303139.

Table[Select[IntegerPartitions[n],UnsameQ@@#&&!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,20}]

A328335 Numbers whose consecutive prime indices are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88

Offset: 1

Gus Wiseman, Oct 14 2019

nonn

First differs from A302569 in having 105, which has 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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of partitions whose consecutive parts are relatively prime (A328172).

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}

A superset of A302569.
Numbers whose prime indices are relatively prime are A289509.
Numbers with no consecutive prime indices relatively prime are A328336.
Cf. A000837, A056239, A112798, A281116, A289508, A318981, A328168, A328169, A328172, A328187, A328188, A328220.

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

A328514 MM-numbers of connected sets of sets.

Original entry on oeis.org

1, 2, 3, 5, 11, 13, 17, 29, 31, 39, 41, 43, 47, 59, 65, 67, 73, 79, 83, 87, 101, 109, 113, 127, 129, 137, 139, 149, 157, 163, 167, 179, 181, 191, 195, 199, 211, 233, 235, 237, 241, 257, 269, 271, 277, 283, 293, 303, 313, 317, 319, 331, 339, 347, 349, 353, 365

Offset: 1

Gus Wiseman, Oct 20 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 all connected set of sets together with their MM-numbers begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
  11: {{3}}
  13: {{1,2}}
  17: {{4}}
  29: {{1,3}}
  31: {{5}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  47: {{2,3}}
  59: {{7}}
  65: {{2},{1,2}}
  67: {{8}}
  73: {{2,4}}
  79: {{1,5}}
  83: {{9}}
  87: {{1},{1,3}}

The not-necessarily-connected case is A302494.
BII-numbers of connected set-systems are A326749.
MM-numbers of connected sets of multisets are A328513.
Cf. A005117, A007947, A056239, A112798, A286518, A302242, A302569, A304714, A305078, A305079, A327398.

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]]]]]]]]];
Select[Range[1000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&]

Intersection of A302494 and A305078.

A337562 Number of pairwise coprime strict compositions of n, where a singleton is always considered coprime.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 7, 17, 13, 23, 41, 41, 67, 49, 75, 75, 155, 211, 229, 243, 241, 287, 395, 807, 537, 841, 655, 1147, 1619, 2037, 2551, 2213, 2007, 2663, 4579, 4171, 7123, 4843, 6013, 6215, 11639, 13561, 16489, 14739, 15445, 16529, 25007, 41003, 32803

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(1) = 1 through a(9) = 12 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)      (9)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)    (1,8)
            (2,1)  (3,1)  (2,3)  (5,1)    (2,5)  (3,5)    (2,7)
                          (3,2)  (1,2,3)  (3,4)  (5,3)    (4,5)
                          (4,1)  (1,3,2)  (4,3)  (7,1)    (5,4)
                                 (2,1,3)  (5,2)  (1,2,5)  (7,2)
                                 (2,3,1)  (6,1)  (1,3,4)  (8,1)
                                 (3,1,2)         (1,4,3)  (1,3,5)
                                 (3,2,1)         (1,5,2)  (1,5,3)
                                                 (2,1,5)  (3,1,5)
                                                 (2,5,1)  (3,5,1)
                                                 (3,1,4)  (5,1,3)
                                                 (3,4,1)  (5,3,1)
                                                 (4,1,3)
                                                 (4,3,1)
                                                 (5,1,2)
                                                 (5,2,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600

A007360 is the unordered version, with non-strict version A051424.
A101268 is the not necessarily strict version.
A220377*6 counts these compositions of length 3.
A337561 does not consider a singleton to be coprime unless it is (1), with non-strict version A337462.
A337664 looks only at distinct parts.
A000740 counts relatively prime compositions, with strict case A332004.
A072706 counts unimodal strict compositions.
A178472 counts compositions with a common factor.
A327516 counts pairwise coprime partitions, with strict case A305713.
A328673 counts pairwise non-coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A087087, A220377, A302569, A307719, A326675, A333227, A335235, A335238, A337461, A337665.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&(Length[#]<=1||CoprimeQ@@#)&]],{n,0,10}]

a(n > 1) = A337561(n) + 1 for n > 1.

A302568 Odd numbers that are either prime or whose prime indices are pairwise coprime.

Original entry on oeis.org

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, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173, 177, 179

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

Also Heinz numbers of partitions with pairwise coprime parts all greater than 1 (A007359), where singletons are considered coprime. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      3: {2}       43: {14}      89: {24}      141: {2,15}
      5: {3}       47: {15}      93: {2,11}    143: {5,6}
      7: {4}       51: {2,7}     95: {3,8}     145: {3,10}
     11: {5}       53: {16}      97: {25}      149: {35}
     13: {6}       55: {3,5}    101: {26}      151: {36}
     15: {2,3}     59: {17}     103: {27}      155: {3,11}
     17: {7}       61: {18}     107: {28}      157: {37}
     19: {8}       67: {19}     109: {29}      161: {4,9}
     23: {9}       69: {2,9}    113: {30}      163: {38}
     29: {10}      71: {20}     119: {4,7}     165: {2,3,5}
     31: {11}      73: {21}     123: {2,13}    167: {39}
     33: {2,5}     77: {4,5}    127: {31}      173: {40}
     35: {3,4}     79: {22}     131: {32}      177: {2,17}
     37: {12}      83: {23}     137: {33}      179: {41}
     41: {13}      85: {3,7}    139: {34}      181: {42}
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
03: {{1}}
05: {{2}}
07: {{1,1}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}
37: {{1,1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
53: {{1,1,1,1}}

A005117 is a superset.
A007359 counts partitions with these Heinz numbers.
A302569 allows evens, with squarefree version A302798.
A337694 is the pairwise non-coprime instead of pairwise coprime version.
A337984 does not include the primes.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions, ranked by A302696.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A005408, A051424, A056239, A087087, A112798, A200976, A302797, A303282, A304711, A335235, A338468.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,400,2],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]

Equals A065091 \/ A337984.

Equals A302569 /\ A005408.

Extended by Gus Wiseman, Oct 29 2020

A337665 Number of compositions of n whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 1, 1, 3, 6, 15, 27, 57, 108, 208, 393, 749, 1415, 2687, 5076, 9583, 18088, 34156, 64511, 121898, 230368, 435460, 823376, 1557420, 2946931, 5578109, 10561987, 20005126, 37902509, 71832372, 136173266, 258211602, 489738622, 929074445, 1762899107, 3345713031

Offset: 0

Gus Wiseman, Sep 22 2020

nonn

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

			The a(1) = 1 through a(5) = 15 compositions:
  (1)  (1,1)  (1,2)    (1,3)      (1,4)
              (2,1)    (3,1)      (2,3)
              (1,1,1)  (1,1,2)    (3,2)
                       (1,2,1)    (4,1)
                       (2,1,1)    (1,1,3)
                       (1,1,1,1)  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220

A000740 is a relatively prime instead of pairwise coprime version.
A304709 is the unordered version.
A333228 ranks these compositions.
A337561 is the strict case.
A337603 is the length-3 case.
A337664 considers all singletons to be coprime.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A337461 counts pairwise coprime length-3 compositions.
Cf. A007359, A007360, A302569, A302696, A304712, A335235, A335238, A337562, A337600, A337601, A337602, A337695.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],CoprimeQ@@Union[#]&]],{n,0,15}]

a(26)-a(35) from Alois P. Heinz, Sep 29 2020

A337694 Numbers with no two relatively prime prime indices.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169, 171, 173, 179, 181, 183, 185, 189, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Sep 23 2020

nonn

First differs from A305078 in having 1 and lacking 195.
First differs from A305103 in having 1 and 169 and lacking 195.
First differs from A328336 in lacking 897, with prime indices (2,6,9).
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.
Also Heinz numbers of integer partitions in which no two parts are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   1: {}      37: {12}     79: {22}      121: {5,5}
   3: {2}     39: {2,6}    81: {2,2,2,2} 125: {3,3,3}
   5: {3}     41: {13}     83: {23}      127: {31}
   7: {4}     43: {14}     87: {2,10}    129: {2,14}
   9: {2,2}   47: {15}     89: {24}      131: {32}
  11: {5}     49: {4,4}    91: {4,6}     133: {4,8}
  13: {6}     53: {16}     97: {25}      137: {33}
  17: {7}     57: {2,8}   101: {26}      139: {34}
  19: {8}     59: {17}    103: {27}      147: {2,4,4}
  21: {2,4}   61: {18}    107: {28}      149: {35}
  23: {9}     63: {2,2,4} 109: {29}      151: {36}
  25: {3,3}   65: {3,6}   111: {2,12}    157: {37}
  27: {2,2,2} 67: {19}    113: {30}      159: {2,16}
  29: {10}    71: {20}    115: {3,9}     163: {38}
  31: {11}    73: {21}    117: {2,2,6}   167: {39}

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

A200976 and A328673 count these partitions.
A302696 and A302569 are pairwise coprime instead of pairwise non-coprime.
A318719 is the squarefree case.
A328867 looks at distinct prime indices.
A337666 is the version for standard compositions.
A101268 counts pairwise coprime or singleton compositions.
A318717 counts strict pairwise non-coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337667 counts pairwise non-coprime compositions.
Cf. A051185, A051424, A056239, A112798, A220377, A284825, A302797, A303282, A305843, A319752, A336737, A337599, A337604, A337605.

filter:= proc(n) local F,i,j,np;
  if n::even and n>2 then return false fi;
  F:= map(t -> numtheory:-pi(t[1]), ifactors(n)[2]);
  np:= nops(F);
  for i from 1 to np-1 do
    for j from i+1 to np do
      if igcd(F[i],F[j])=1 then return false fi
  od od;
  true
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 06 2020

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Select[Range[100],stabQ[primeMS[#],CoprimeQ]&]

A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

Original entry on oeis.org

18, 36, 42, 45, 50, 54, 72, 75, 78, 84, 90, 98, 99, 100, 105, 108, 114, 126, 130, 135, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 195, 196, 198, 200, 207, 210, 216, 222, 225, 228, 230, 231, 234, 242, 245, 250, 252, 258, 260, 266, 270, 275, 279, 285, 288

Offset: 1

Gus Wiseman, Apr 20 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of integer partitions whose Heinz numbers belong to this sequence begins (221), (2211), (421), (322), (331), (2221), (22111), (332), (621), (4211), (3221), (441), (522), (3311), (432), (22211).

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303283.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

A304712 Number of integer partitions of n whose parts are all equal or whose distinct parts are pairwise coprime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 19, 25, 32, 43, 54, 70, 86, 105, 130, 162, 196, 240, 286, 339, 405, 485, 573, 674, 790, 922, 1072, 1252, 1456, 1685, 1939, 2226, 2557, 2923, 3349, 3822, 4347, 4931, 5593, 6335, 7170, 8092, 9105, 10233, 11495, 12903, 14458, 16169, 18063

Offset: 0

Gus Wiseman, May 17 2018

nonn

Two parts are coprime if they have no common divisor greater than 1.

			The a(6) = 10 partitions whose parts are all equal or whose distinct parts are pairwise coprime are (6), (51), (411), (33), (321), (3111), (222), (2211), (21111), (111111).

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000837, A007359, A018783, A051424, A056239, A078374, A101268, A289508, A289509, A298748, A300486, A302569, A302696, A302698, A302796, A302797, A304709, A304711.

g:= proc(n, i, s) `if`(n=0, 1, `if`(i<1, 0,
      b(n, i, select(x-> x<=i, s))))
    end:
b:= proc(n, i, s) option remember; g(n, i-1, s)+(f->
     `if`(f intersect s={}, add(g(n-i*j, i-1, s union f)
        , j=1..n/i), 0))(numtheory[factorset](i))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2018

Table[Select[IntegerPartitions[n],Or[SameQ@@#,CoprimeQ@@Union[#]]&]//Length,{n,20}]
(* Second program: *)
g[n_, i_, s_] := If[n == 0, 1, If[i < 1, 0, b[n, i, Select[s, # <= i &]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + Function[f,
     If[f ~Intersection~ s == {}, Sum[g[n - i*j, i - 1, s ~Union~ f],
     {j, 1, n/i}], 0]][FactorInteger[i][[All, 1]]];
a[n_] := g[n, n, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

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