A327516 -id:A327516 - cp's OEIS frontend

A336735 Products of elements of A304711.

1, 2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106

Offset: 1

Gus Wiseman, Aug 02 2020

nonn

A304711 lists numbers whose distinct prime indices are pairwise coprime.

First differs from A304711 in having 84.

			The sequence of terms together with their prime indices begins:
      1: {}            28: {1,1,4}         52: {1,1,6}
      2: {1}           30: {1,2,3}         54: {1,2,2,2}
      4: {1,1}         32: {1,1,1,1,1}     55: {3,5}
      6: {1,2}         33: {2,5}           56: {1,1,1,4}
      8: {1,1,1}       34: {1,7}           58: {1,10}
     10: {1,3}         35: {3,4}           60: {1,1,2,3}
     12: {1,1,2}       36: {1,1,2,2}       62: {1,11}
     14: {1,4}         38: {1,8}           64: {1,1,1,1,1,1}
     15: {2,3}         40: {1,1,1,3}       66: {1,2,5}
     16: {1,1,1,1}     44: {1,1,5}         68: {1,1,7}
     18: {1,2,2}       45: {2,2,3}         69: {2,9}
     20: {1,1,3}       46: {1,9}           70: {1,3,4}
     22: {1,5}         48: {1,1,1,1,2}     72: {1,1,1,2,2}
     24: {1,1,1,2}     50: {1,3,3}         74: {1,12}
     26: {1,6}         51: {2,7}           75: {2,3,3}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

A181818 is the version for superprimorials, with complement A336426.
A336496 is the version for superfactorials, with complement A336497.
A336620 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A001221, A007360, A007916, A056239, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A336424, A336497, A336736.

nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]!={}&]

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

A336735 Products of elements of A304711.

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