A337694 Numbers with no two relatively prime prime indices.

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

1,2

Comments

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).

Links

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

Example

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}

Maple

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

Mathematica

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]&]

Crossrefs

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.

Sequence in context: A229511 A305078 A328336 * A305103 A065520 A316428

Adjacent sequences: A337691 A337692 A337693 * A337695 A337696 A337697

Keyword

nonn

Author

Gus Wiseman, Sep 23 2020

Status

approved