A335236 -id:A335236 - cp's OEIS frontend

A335241 Numbers whose prime indices are not pairwise coprime, where a singleton is not coprime unless it is {1}.

1, 3, 5, 7, 9, 11, 13, 17, 18, 19, 21, 23, 25, 27, 29, 31, 36, 37, 39, 41, 42, 43, 45, 47, 49, 50, 53, 54, 57, 59, 61, 63, 65, 67, 71, 72, 73, 75, 78, 79, 81, 83, 84, 87, 89, 90, 91, 97, 98, 99, 100, 101, 103, 105, 107, 108, 109, 111, 113, 114, 115, 117, 121

Offset: 1

Gus Wiseman, May 30 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

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:
    1: {}          31: {11}          61: {18}
    3: {2}         36: {1,1,2,2}     63: {2,2,4}
    5: {3}         37: {12}          65: {3,6}
    7: {4}         39: {2,6}         67: {19}
    9: {2,2}       41: {13}          71: {20}
   11: {5}         42: {1,2,4}       72: {1,1,1,2,2}
   13: {6}         43: {14}          73: {21}
   17: {7}         45: {2,2,3}       75: {2,3,3}
   18: {1,2,2}     47: {15}          78: {1,2,6}
   19: {8}         49: {4,4}         79: {22}
   21: {2,4}       50: {1,3,3}       81: {2,2,2,2}
   23: {9}         53: {16}          83: {23}
   25: {3,3}       54: {1,2,2,2}     84: {1,1,2,4}
   27: {2,2,2}     57: {2,8}         87: {2,10}
   29: {10}        59: {17}          89: {24}

The complement is A302696.
The version for relatively prime instead of coprime is A318978.
The version for standard compositions is A335239.
These are the Heinz numbers of the partitions counted by A335240.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Primes and numbers with pairwise coprime prime indices are A302569.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime standard composition numbers are A333227.
Cf. A007360, A101268, A326674, A327516, A333228, A335236, A335237, A335238.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!CoprimeQ@@primeMS[#]&]

A338318 Composite numbers whose prime indices are pairwise intersecting (non-coprime).

Original entry on oeis.org

9, 21, 25, 27, 39, 49, 57, 63, 65, 81, 87, 91, 111, 115, 117, 121, 125, 129, 133, 147, 159, 169, 171, 183, 185, 189, 203, 213, 235, 237, 243, 247, 259, 261, 267, 273, 289, 299, 301, 303, 305, 319, 321, 325, 333, 339, 343, 351, 361, 365, 371, 377, 387, 393

Offset: 1

Gus Wiseman, Oct 31 2020

nonn

First differs from A322336 in lacking 2535, with prime indices {2,3,6,6}.
First differs from A327685 in having 17719, with prime indices {6,10,15}.
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 pairwise intersecting (non-coprime) partitions with more than one part. 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:
      9: {2,2}        121: {5,5}        243: {2,2,2,2,2}
     21: {2,4}        125: {3,3,3}      247: {6,8}
     25: {3,3}        129: {2,14}       259: {4,12}
     27: {2,2,2}      133: {4,8}        261: {2,2,10}
     39: {2,6}        147: {2,4,4}      267: {2,24}
     49: {4,4}        159: {2,16}       273: {2,4,6}
     57: {2,8}        169: {6,6}        289: {7,7}
     63: {2,2,4}      171: {2,2,8}      299: {6,9}
     65: {3,6}        183: {2,18}       301: {4,14}
     81: {2,2,2,2}    185: {3,12}       303: {2,26}
     87: {2,10}       189: {2,2,2,4}    305: {3,18}
     91: {4,6}        203: {4,10}       319: {5,10}
    111: {2,12}       213: {2,20}       321: {2,28}
    115: {3,9}        235: {3,15}       325: {3,3,6}
    117: {2,2,6}      237: {2,22}       333: {2,2,12}

A200976 counts the partitions with these Heinz numbers.
A302696 is the pairwise coprime instead of pairwise non-coprime version.
A337694 includes the primes.
A002808 lists composite numbers.
A318717 counts pairwise intersecting strict partitions.
A328673 counts partitions with pairwise intersecting distinct parts, with Heinz numbers A328867 and restriction to triples A337599 (except n = 3).
Cf. A008578, A051185, A056239, A101268, A112798, A284825, A302569, A305843, A319752, A327516, A335236, A337666, A337667.

stabstrQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Select[Range[2,100],!PrimeQ[#]&&stabstrQ[PrimePi/@First/@FactorInteger[#],CoprimeQ]&]

Equals A337694 \ A008578.

cp's OEIS Frontend

A335241 Numbers whose prime indices are not pairwise coprime, where a singleton is not coprime unless it is {1}.

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