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

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..54.

Formula

Equals A337694 \ A008578.

Example

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}

Mathematica

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

Crossrefs

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.

Sequence in context: A305733 A139392 A274720 * A322336 A327685 A355535

Adjacent sequences: A338315 A338316 A338317 * A338319 A338320 A338321

Keyword

nonn

Author

Gus Wiseman, Oct 31 2020

Status

approved